?Explain how to determine if two functions are inverses of each other

Fill in each blank so that the resulting statement is true. The notation \(f^{-1}\) means the ____________ of the function \(f\). Equation Transcription: Text Transcription: f^-1 f

Fill in each blank so that the resulting statement is true. If the function \(g\) is the inverse of the function \(f\), then \(f(g(x))=\) ___________ and \(g(f(x))=\) ______________. Equation Transcription: Text Transcription: g f f(g(x)) = g(f(x)) =

Fill in each blank so that the resulting statement is true. A function \(f\) has an inverse that is a function if there is no __________ line that intersects the graph of \(f\) at more than one point. Such a function is called a/an ______________ function. Equation Transcription: Text Transcription: f

Fill in each blank so that the resulting statement is true. The graph of \(f^{-1}\) is a reflection of the graph of \(f\) about the line whose equation is ________________. Equation Transcription: Text Transcription: f^-1 f

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=4 x\) and \(g(x)=\frac{x}{4}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 4x g(x) = x/4

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=6 x\) and \(g(x)=\frac{x}{6}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 6x g(x) = x/6

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=3 x+8\) and \(g(x)=\frac{x-8}{3}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 3x + 8 g(x) = x-8 / 3

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=4 x+9\) and \(g(x)=\frac{x-9}{4}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 4x + 9 g(x) = x-9 / 4

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=5 x-9\) and \(g(x)=\frac{x+5}{9}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 5x - 9 g(x) = x+5 /9

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=3 x-7\) and \(g(x)=\frac{x+3}{7}\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 3x - 7 g(x) = x+3 /7

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=\frac{3}{x-4}\) and \(g(x)=\frac{3}{x}+4\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 3 / x - 4 g(x) = 3 /x + 4

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=\frac{2}{x-5}\) and \(g(x)=\frac{2}{x}+5\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = 2 / x - 5 g(x) = 2/x + 5

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=-x\) and \(g(x)=-x\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = -x g(x) = -x

In Exercises \(1–10\), find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\) Equation Transcription: Text Transcription: 1-10 f(g(x)) g(f(x)) f g f(x) = cube root x - 4 g(x) = x^3 + 4

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x+3\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = x + 3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x+5\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = x + 5

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=2 x\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 2x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=4 x\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 4x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=2 x+3\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 2x + 3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=3 x-1\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 3x - 1

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x^{3}+2\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = x^3 + 2

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x^{3}-1\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = x^3 - 1

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=(x+2)^{3}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = (x + 2)^3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=(x-1)^{3}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = (x - 1)^3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{1}{x}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 1 / x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{2}{x}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 2 / x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\sqrt{x}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = sqrt x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\sqrt[3]{x}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = cube root x

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{7}{x}-3\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 7 / x - 3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{4}{x}+9\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 4 / x + 9

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{2 x+1}{x-3}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 2x + 1 / x - 3

The functions in Exercises \(11–28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x)\), the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=\frac{2 x-3}{x+1}\) Equation Transcription: Text Transcription: 11-28 f^-1(x) f(f^-1(x)) = x f^-1(f(x)) = x f(x) = 2x - 3 / x + 1

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

Which graphs in Exercises \(29–34\) represent functions that have inverse functions? Equation Transcription: Text Transcription: 29-34

In Exercises \(35–38\), use the graph of \(f\) to draw the graph of its inverse function. Equation Transcription: Text Transcription: 35-38 f

In Exercises \(35–38\), use the graph of \(f\) to draw the graph of its inverse function. Equation Transcription: Text Transcription: 35-38 f

In Exercises \(35–38\), use the graph of \(f\) to draw the graph of its inverse function. Equation Transcription: Text Transcription: 35-38 f

In Exercises \(35–38\), use the graph of \(f\) to draw the graph of its inverse function. Equation Transcription: Text Transcription: 35-38 f

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=2 x-1\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = 2x - 1

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=2 x-3\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = 2x - 3

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=x^{2}-4, x \geq 0\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = x^2 - 4, x geq 0

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=x^{2}-1, x \leq 0\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = x^2 - 1, x leq 0

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=(x-1)^{2}, x \leq 1\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = (x - 1)^2 , x leq 1

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=(x-1)^{2}, x \geq 1\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = (x - 1)^2 , x geq 1

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=x^{3}-1\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = x^3 - 1

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=x^{3}+1\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = x^3 + 1

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=(x+2)^{3}\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = (x + 2)^3

In Exercises \(39–52\), a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). \(f(x)=(x-2)^{3}\) Equation Transcription: Text Transcription: 39-52 f^-1(x) f f^-1 f(x) = (x - 2)^3

(Hint for Exercises \(49–52\): To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y\) .) \(f(x)=\sqrt{x-1}\) Equation Transcription: Text Transcription: 49-52 (n root y)^n = y f(x) = sqrt x - 1

(Hint for Exercises \(49–52\): To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y\) .) \(f(x)=\sqrt{x+2}\) Equation Transcription: Text Transcription: 49-52 (n root y)^n = y f(x) = sqrt x + 2

(Hint for Exercises \(49–52\): To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y\) .) \(f(x)=\sqrt[3]{x}+1\) Equation Transcription: Text Transcription: 49-52 (n root y)^n = y f(x) = cube root x + 1

(Hint for Exercises \(49–52\): To solve for a variable involving an nth root, raise both sides of the equation to the nth power: \((\sqrt[n]{y})^{n}=y\) .) \(f(x)=\sqrt[3]{x-1}\) Equation Transcription: Text Transcription: 49-52 (n root y)^n = y f(x) = cube root x - 1

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \(f(g(1))\) Equation Transcription: Text Transcription: 53-58 f g f(g(1))

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \(f(g(4))\) Equation Transcription: Text Transcription: 53-58 f g f(g(4))

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \((g \circ f)(-1)\) Equation Transcription: Text Transcription: 53-58 f g (g circ f)(-1)

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \((g \circ f)(0)\) Equation Transcription: Text Transcription: 53-58 f g (g circ f)(0)

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \(f^{-1}(g(10))\) Equation Transcription: Text Transcription: 53-58 f g f ^-1 (g(10))

In Exercises \(53–58\), \(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. \(f^{-1}(g(1))\) Equation Transcription: Text Transcription: 53-58 f g f ^-1 (g(1))

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \((f \circ g)(0)\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 (f circ g)(0)

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \((g \circ f)(0)\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 (g circ f)(0)

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \(f^{-1}(1)\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 f^-1 (1)

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \(g^{-1}(7)\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 g^-1 (7)

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \(g(f[h(1)])\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 g(f[h(1)])

In Exercises \(59–64\), let \(f(x)=2 x-5\) \(g(x)=4 x-1\) \(h(x)=x^{2}+x+2\). Evaluate the indicated function without finding an equation for the function. \(f(g[h(1)])\) Equation Transcription: Text Transcription: 59-64 f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x^2 + x + 2 f(g[h(1)])

In most societies, women say they prefer to marry men who are older than themselves, whereas men say they prefer women who are younger. The bar graph shows the preferred age difference in a mate in five selected countries. Use this information to solve Exercises \(65–66\). a. Consider a function, \(f\), whose domain is the set of the five countries shown in the graph. Let the range be the set of the average number of years women in each of the respective countries prefer men who are older than themselves. Write the function \(f\) as a set of ordered pairs. b. Write the relation that is the inverse of \(f\) as a set of ordered pairs. Based on these ordered pairs, is \(f\) a one- to-one function? Explain your answer. Equation Transcription: Text Transcription: 65-66 f

In most societies, women say they prefer to marry men who are older than themselves, whereas men say they prefer women who are younger. The bar graph shows the preferred age difference in a mate in five selected countries. Use this information to solve Exercises \(65–66\). a. Consider a function, \(g\), whose domain is the set of the five countries shown in the graph. Let the range be the set of negative numbers representing the average number of years men in each of the respective countries prefer women who are younger than themselves. Write the function \(g\) as a set of ordered pairs. b. Write the relation that is the inverse of \(g\) as a set of ordered pairs. Based on these ordered pairs, is \(g\) a one-to-one function? Explain your answer. Equation Transcription: Text Transcription: 65-66 g

The graph represents the probability of two people in the same room sharing a birthday as a function of the number of people in the room. Call the function \(f\). a. Explain why \(f\) has an inverse that is a function. b. Describe in practical terms the meaning of \(f^{-1}(0.25), f^{-1}(0.5), \text { and } f^{-1}(0.7)\). Equation Transcription: Text Transcription: f f^-1(0.25), f^-1(0.5), and f^-1(0.7)

A study of \(900\) working women in Texas showed that their feelings changed throughout the day. As the graph indicates, the women felt better as time passed, except for a blip (that’s slang for relative maximum) at lunchtime. a. Does the graph have an inverse that is a function? Explain your answer. b. Identify two or more times of day when the average happiness level is \(3\). Express your answers as ordered pairs. c. Do the ordered pairs in part (b) indicate that the graph represents a one-to-one function? Explain your answer. Equation Transcription: Text Transcription: 900 3

The formula \(y=f(x)=\frac{9}{5} x+32\) is used to convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit. The formula \(y=g(x)=\frac{5}{9}(x-32)\) is used to convert from \(x\) degrees Fahrenheit to \(y\) degrees Celsius. Show that \(f\) and \(g\) are inverse functions. Equation Transcription: Text Transcription: y = f(x) =9/5 x + 32 x y y = g(x) = 5/9 (x - 32) f g

Explain how to determine if two functions are inverses of each other.

Describe how to find the inverse of a one-to-one function.

What is the horizontal line test and what does it indicate?

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

How can a graphing utility be used to visually determine if two functions are inverses of each other?

What explanations can you offer for the trends shown by the graph in Exercise \(68\)? Equation Transcription: Text Transcription: 68

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=x^{2}-1\) Equation Transcription: Text Transcription: 76-83 f(x) = x^2 - 1

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=\sqrt[3]{2-x}\) Equation Transcription: Text Transcription: 76-83 f(x) = cube root 2 - x

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=\frac{x^{3}}{2}\) Equation Transcription: Text Transcription: 76-83 f(x) = x^3 / 2

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=\frac{x^{4}}{4}\) Equation Transcription: Text Transcription: 76-83 f(x) = x^4 / 4

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=\operatorname{int}(x-2)\) Equation Transcription: Text Transcription: 76-83 f(x) = int(x - 2)

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=|x-2|\) Equation Transcription: Text Transcription: 76-83 f(x) = |x - 2|

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=(x-1)^{3}\) Equation Transcription: Text Transcription: 76-83 f(x) = (x - 1)^3

In Exercises \(76–83\), use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). \(f(x)=-\sqrt{16-x^{2}}\) Equation Transcription: Text Transcription: 76-83 f(x) = - sqrt 16 - x^2

In Exercises \(84–86\), use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and \(g\) are inverses \(f(x)=4 x+4, g(x)=0.25 x-1\) Equation Transcription: Text Transcription: 84-86 f g y = x f(x) = 4x + 4, g(x) = 0.25x - 1

In Exercises \(84–86\), use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and \(g\) are inverses \(f(x)=\frac{1}{x}+2, g(x)=\frac{1}{x-2}\) Equation Transcription: Text Transcription: 84-86 f g y = x f(x) = 1 / x + 2, g(x) = 1 / x - 2

In Exercises \(84–86\), use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and \(g\) are inverses \(f(x)=\sqrt[3]{x}-2, g(x)=(x+2)^{3}\) Equation Transcription: Text Transcription: 84-86 f g y = x f(x) = cube root x - 2, g(x) = (x + 2)^3

In Exercises \(87–90\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I found the inverse of \(f(x)=5 x-4\) in my head: The reverse of multiplying by \(5\) and subtracting \(4\) is adding \(4\) and dividing by \(5\), so \(f^{-1}(x)=\frac{x+4}{5}\). Equation Transcription: Text Transcription: 87-90 f(x) = 5x - 4 5 4 f^-1(x) = x + 4 / 5

In Exercises \(87–90\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I’m working with the linear function \(f(x)=3 x+5\) and I do not need to find \(f^{-1}\) in order to determine the value of \(\left.f \circ f^{-1}\right)(17)\). Equation Transcription: Text Transcription: 87-90 f(x) = 3x + 5 f ^-1 (f circ f^ -1 )(17)

In Exercises \(87–90\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Exercises \(89–90\) are based on the following cartoon. Assuming that there is no such thing as metric crickets, I modeled the information in the first frame of the cartoon with the function \(T(n)=\frac{y}{4}+40\), where \(T(n)\) is the temperature, in degrees Fahrenheit, and \(n\) is the number of cricket chirps per minute. Equation Transcription: Text Transcription: 87-90 89-90 T(n) = n /4 + 40 T(n) n

In Exercises \(87–90\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Exercises \(89–90\) are based on the following cartoon. I used the function in Exercise \(89\) and found an equation for \(T^{-1}\), which expresses the number of cricket chirps per minute as a function of Fahrenheit temperature. Equation Transcription: Text Transcription: 87-90 89-90 89 T^-1

In Exercises \(91–94\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inverse of \(\{(1,4),(2,7)\}\) is \(\{(2,7),(1,4)\}\). Equation Transcription: Text Transcription: 91-94 {(1,4), (2,7)} {(2,7), (1,4)}

In Exercises \(91–94\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The function \(f(x)=5\) is one-to-one. Equation Transcription: Text Transcription: 91-94 f(x) = 5

In Exercises \(91–94\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=3 x\), then \(f^{-1}(x)=\frac{1}{3 x}\). Equation Transcription: Text Transcription: 91-94 f(x) = 3x f^ -1 (x) = 1 / 3x

In Exercises \(91–94\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The domain of \(f\) is the same as the range of \(f^{-1}\). Equation Transcription: Text Transcription: 91-94 f f^-1

If \(f(x)=3 x\) and \(g(x)=x+5\), find \((f \circ g)^{-1}(x)\) and \(\left(g^{-1} \circ f^{-1}\right)(x)\). Equation Transcription: Text Transcription: f(x) = 3x g(x) = x + 5 (f circ g)^-1(x) (g^-1 circ f^-1)(x)

Show that \(f(x)=\frac{3 x-2}{5 x-3}\) is its own inverse. Equation Transcription: Text Transcription: f(x) = 3x - 2 / 5x - 3

Freedom \(7\) was the spacecraft that carried the first American into space in \(1961\). Total flight time was \(15\) minutes and the spacecraft reached a maximum height of \(116\) miles. Consider a function, \(s\), that expresses Freedom \(7\) ’s height, \(s(t)\), in miles, after \(t\) minutes. Is \(s\) a one-to-one function? Explain your answer. Equation Transcription: Text Transcription: 7 1961 15 116 s s(t) t

If \(f(2)=6\), and \(f\) is one-to-one, find \(x\) satisfying \(8+f^{-1}(x-1)=10\). Equation Transcription: Text Transcription: f(2) = 6 f x 8 + f^-1(x-1)=10

In Tom Stoppard’s play Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.

Exercises \(100–102\) will help you prepare for the material covered in the next section. Let \(\left(x_{1}, y_{1}\right)=(7,2)\) and \(\left(x_{2}, y_{2}\right)=(1,-1)\). Find \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\) . Express the answer in simplified radical form. Equation Transcription: Text Transcription: 100-102 (x_1,y_1) =(7,2) (x_2,y_2) =(1,-1) sqrt (x_2 - x_1)^2 + (y_2 - y_1)^2

Exercises \(100–102\) will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center \((1, -1)\) and radius \(1\). Equation Transcription: Text Transcription: 100-102 (1, -1) 1

Exercises \(100–102\) will help you prepare for the material covered in the next section. Solve by completing the square: \(y^{2}-6 y-4=0\). Equation Transcription: Text Transcription: 100-102 y^2 - 6y - 4 = 0