524 Sketch the graph of the equation by translating, reflecting, compressing, and

Let \(f(x)=3 \sqrt{x}-2\) and \(g(x)=|x|\). In each part, give the formula for the function and state the corresponding domain. (a) \(f+g:\) ________ Domain: ________ (b) \(f-g:\) ________ Domain: ________ (c) \(\text { fg : }\) ________ Domain: ________ (d) \(f / g:\)_______ Domain: ________ Equation Transcription: _____ _____ _____ _____ Text Transcription: f(x)=3 sqrt x-2 g(x)=|x| f+g:_____ f-g:_____ fg:_____ f/g:____

Let \(f(x)=2-x^{2}\) and \(g(x)=\sqrt{x}\). In each part, give the formula for the composition and state the corresponding domain. (a) \(f \circ g:\) ________ Domain: ________ (b) \(g \circ f:\) ________ Domain: ________ Equation Transcription: ______ ______ Text Transcription: f(x)=2-x^2 g(x)=x f circ g:______ g circ f:______

The graph of \(y=1+(x-2)^{2}\) may be obtained by shifting the graph of \(y=x^{2}\) _______ (left/right) by _______ unit(s) and then shifting this new graph _______ (up/down) by _______ unit(s). Equation Transcription: ______ Text Transcription: y=1+(x-2)^2 y=x^2

Let \(f(x)=\left\{\begin{array}{lr}|x+1|, & -2 \leq x \leq 0 \\|x-1|, & 0<x \leq 2\end{array}\right.\) (a) The letter of the alphabet that most resembles the graph of \(f\) is ________. (b) Is \(f\) an even function? Equation Transcription: { Text Transcription: f(x)={_|x-1|,0 < x leq 2 ^|x + 1|, -2 leq x leq 0 f f

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=-2(x+)^{2}-3\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=-2(x+)^2-3

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{1}{2}(x-3)^{2}+2\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1/2(x-3)^2+2

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=x^{2}+6 x\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=x^2+6x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{1}{2}\left(x^{2}-2 x+3\right)\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1/2(x^2-2x+3)

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=3-\sqrt{x+1}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=3-sqrt x+1

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=1+\sqrt{x-4}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1+sqrt x-4

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{1}{2} \sqrt{x}+1\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1/2 sqrt x+1

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=-\sqrt{3 x}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=-sqrt3x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{1}{x-3}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1/x - 3

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{1}{1-x}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1/1 - x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=2-\frac{1}{1+x}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=2-1/1 + x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\frac{x-1}{x}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=x -1/x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=|x+2|-2\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=|x+2|-2

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=1-|x-3|\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1-|x-3|

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=|2 x-1|+1\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=|2x-1|+1

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\sqrt{x^{2}-4 x+4}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=sqrt x^2-4x +4

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=1-2 \sqrt[3]{x}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=1-2 sqrt ^3 _x

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\sqrt[3]{x-2-3}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=sqrt ^3 _x-2-3

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=2+\sqrt[3]{x+1}\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=2+sqrt ^3 _x+1

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}\), \(y=\sqrt{x}\), \(y=1 / x\), \(y=|x|\), or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. \(y=\sqrt[3]{x-2}=0\) Equation Transcription: or Text Transcription: y=x^2, y=x, y=1/x, y=|x|, or y= sqrt^3_x y=sqrt^3 _x-2=0

(a) Sketch the graph of \(y=x+|x|\) by adding the corresponding \(y \text { - coordinates }\) on the graphs of \(y=x \)and \(y=|x|\). (b) Express the equation \(y=x+|x|\) in piecewise form with no absolute values, and confirm that the graph you obtained in part (a) is consistent with this equation. Equation Transcription: Text Transcription: y=x+|x| y-coordinates y=x y=|x| y=x+|x|

Sketch the graph of \(y=x+(1 / x)\) by adding corresponding \(y \text {-coordinates }\) on the graphs of \(y=x\) and \(y=1 / x\). Use a graphing utility to confirm that your sketch is correct. Equation Transcription: Text Transcription: y=x+(1/x) y-coordinates y=x y=1/x

Find formulas for \(f+g\), \(f-g\), \(f g\) and \(f / g\), and state the domains of the functions. \(f(x)=2 \sqrt{x-1}, g(x)=\sqrt{x-1}\) Equation Transcription: and Text Transcription: f+g,f-g,fg and f/g f(x)=2 sqrt x-1, g(x)=sqrt x-1

Find formulas for \(f+g\), \(f-g\), \(f g\) and \(f / g\), and state the domains of the functions. \(f(x)=\frac{x}{1+x^{2}}, g(x)=\frac{1}{x}\) Equation Transcription: and Text Transcription: f+g,f-g,fg and f/g f(x)=x/1 + x^2,g(x)=1/x

Let \(f(x)=\sqrt{x}\) and \(g(x)=x^{3}+1\). Find (a) \(f(g(2))\) (b) \(g(f(4))\) (c) \(f(f(16))\) (d) \(g(g(0))\) (e) \(f(2+h)\) (f ) \(g(3+h)\). Equation Transcription: Text Transcription: f(x)=sqrt x g(x)=x^3+1 f(g(2)) g(f(4)) f(f(16)) g(g(0)) f(2+h) g(3+h)

Let \(g(x)=\sqrt{x}\). Find (a) \(g(5 s+2)\) (b) \(g(\sqrt{x}+2)\) (c) \(3 g(5 x)\) (d) \(\frac{1}{g(x)}\) (e) \(g(g(x))\) (f) \((g(x))^{2}-g\left(x^{2}\right)\) (g) \(g(1 / \sqrt{x})\) (h) \(g\left((x-1)^{2}\right)\) (i) \(g(x+h)\). Equation Transcription: Text Transcription: g(x)=sqrt x g(5s+2) g(sqrt x+2) 3g(5x) 1/g(x) g(g(x)) (g(x))^2-g(x^2) g(1/sqrt x) g((x-1)^2) g(x+h)

Find formulas for \(f \circ g\) and \(g \circ f\) , and state the domains of the compositions. \(f(x)=x^{2}, g(x)=\sqrt{1-x}\) Equation Transcription: Text Transcription: F circ g G circ f f(x)=x^2, g(x)=sqrt 1-x

Find formulas for \(f \circ g\) and \(g \circ f\) , and state the domains of the compositions. \(f(x)=\sqrt{x-3}, g(x)=\sqrt{x^{2}+3}\) Equation Transcription: Text Transcription: F circ g G circ f f(x)=sqrt x-3, g(x)=sqrt x^2+3

Find formulas for \(f \circ g\) and \(g \circ f\) , and state the domains of the compositions. \(f(x)=\frac{1+x}{1-x}, g(x)=\frac{x}{1-x}\) Equation Transcription: Text Transcription: F circ g G circ f f(x)=1 + x/1 - x, g(x)=x/1 - x

Find formulas for \(f \circ g\) and \(g \circ f\) , and state the domains of the compositions. \(f(x)=\frac{1}{1+x^{2}}, g(x)=\frac{1}{x}\) Equation Transcription: Text Transcription: F circ g G circ f f(x)=1/1 + x^2, g(x)=1/x

Find a formula for \(f \circ g \circ h\). \(f(x)=x^{2}+1, g(x)=\frac{1}{x}, h(x)=x^{3}\) Equation Transcription: Text Transcription: F circ g circ h f(x)=x^2+1, g(x)=1/x, h(x)=x^3

Find a formula for \(f^{\circ} g^{\circ} h\) \(f(x)=\frac{1}{1=x_{t}} g(x)=\sqrt[3]{x,} h(x)=\frac{1}{x^{3}}\) Equation Transcription: Text Transcription: F circ g circ h f(x) = frac{1}{1 = x_{t}} g(x) = sqrt[3]{x,} h(x) = frac{1}{x^{3}}

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a)\(f(x)=\sqrt{x+2}\) (b) \(f(x)=\left|x^{2}-3 x+5\right|\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = sqrt{x+2} f(x) = |x^{2}-3 x+5|

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a) \(f(x)=x^{2}+1\) (b) \(f(x)=\frac{1}{x-3}\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = x^{2}+1 f(x) = frac{1}{x-3}

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a) \(f(x)=\sin ^{2} x\) (b) \(f(x)=\frac{3}{5 \cos x}\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = sin ^{2} x f(x) = frac{3}{5 cos x}

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a) \(f(x)=3 \sin \left(x^{2}\right)\) (b) \(f(x)=3 \sin ^{2} x+\sin x\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = 3 sin (x^2) f(x) = 3 sin ^2 x + sin x

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a) \(f(x)=\left(1+\sin \left(x^{2}\right)\right)^{3}\) (b) \(f(x)=\sqrt{1-\sqrt[3]{x}}\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = (1+sin (x^{2}))^3 f(x) = sqrt{1 - sqrt[3]{x}}

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h\). [Note: Each exercise has more than one solution.] (a) \(f(x)=\frac{1}{1-x^{2}}\) (b) \(f(x)=|5+2 x|\) Equation Transcription: Text Transcription: f g h f = g circ h f(x) = frac{1}{1-x^2} f(x) = |5+2 x|

True-False Determine whether the statement is true or false. Explain your answer. The domain of \(f + g\) is the intersection of the domains of \(f\) and \(g\). Equation Transcription: Text Transcription: f + g f g

True-False Determine whether the statement is true or false. Explain your answer. The domain of \(f \circ g\) consists of all values of \(x\) in the domain of \(g\) for which \(g \neq 0\) Equation Transcription: Text Transcription: f circ g x g f neq 0

True-False Determine whether the statement is true or false. Explain your answer. The graph of an even function is symmetric about the \(y-\text { axis }\). Equation Transcription: Text Transcription: y - axis

True-False Determine whether the statement is true or false. Explain your answer. The graph of \(y=f(x+2)+3\) is obtained by translating the graph of \(y=f(x)\) right 2 units and up 3 units. Equation Transcription: Text Transcription: y = f(x + 2) + 3 y = f(x)

Use the data in the accompanying table to make a plot of \(y=f(g(x))\). Equation Transcription: Text Transcription: y = f(g(x))

Find the domain of \(g \circ f\) for the functions \(f\) and \(g\) in Exercise 47 Equation Transcription: Text Transcription: g circ f f g

Sketch the graph of \(y=f(g(x))\)for the functions graphed in the accompanying figure. Equation Transcription: Text Transcription: y = f(g(x))

Sketch the graph of \(y = g(f(x))\) for the functions graphed in Exercise 49. Equation Transcription: Text Transcription: y = g(f(x))

Use the graphs of \(f\) and \(g\) in Exercise 49 to estimate the solutions of the equations \(f(g(x)) = 0\) and \(g(f(x)) = 0\). Equation Transcription: Text Transcription: f g f(g(x)) g(f(x))

Use the table given in Exercise 47 to solve the equations \(f(g(x)) = 0\) and \(g(f(x)) = 0\). Equation Transcription: Text Transcription: f = (g(x)) = 0 g = (f(x)) = 0

Find \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(w)-f(x)}{w-x}\) Simplify as much as possible. \(f(x)=3 x^{2}-5\) Equation Transcription: and Text Transcription: frac{f(x+h)-f(x)}{h} frac{f(w)-f(x)}{w-x} f(x) = 3 x^{2} - 5

Find \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(w)-f(x)}{w-x}\) Simplify as much as possible. \(f(x)=x^{2}+6 x\) Equation Transcription: and Text Transcription: frac{f(x+h)-f(x)}{h} frac{f(w)-f(x)}{w-x} f(x) = x^2 + 6x

Find \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(w)-f(x)}{w-x}\) Simplify as much as possible. \(f(x)=1 / x\) Equation Transcription: and Text Transcription: frac{f(x+h)-f(x)}{h} frac{f(w)-f(x)}{w-x} f(x) = 1/x

Find \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(w)-f(x)}{w-x}\) Simplify as much as possible. \(f(x)=1 / x^{2}\) Equation Transcription: and Text Transcription: frac{f(x+h)-f(x)}{h} frac{f(w)-f(x)}{w-x} f(x) = 1/x^2

Classify the functions whose values are given in the accompanying table as even, odd, or neither.

Complete the accompanying table so that the graph of \( = f(x)\) is symmetric about (a) the \(y\) - axis (b) the origin. Equation Transcription: Text Transcription: y = f(x) y

The accompanying figure shows a portion of a graph. Complete the graph so that the entire graph is symmetric about (a) the \(x\) -axis (b) the \(y\) -axis (c) the origin. Equation Transcription: Text Transcription: x y

The accompanying figure shows a portion of the graph of a function \(f\) . Complete the graph assuming that (a) \(f\) is an even function (b) \(f\) is an odd function. Equation Transcription: Text Transcription: f f f

Classify the functions graphed in the accompanying figures as even, odd, or neither.

Classify the functions graphed in the accompanying figures as even, odd, or neither.

In each part, classify the function as even, odd, or neither. (a) \(f(x)=x^{2}\) (b) \(f(x)=x^{2}\) (c) \(f(x)=|x|\) (d) \(f(x)= x + 1\) (e) \(f(x)=\frac{x^{5}-x}{1+x^{2}}\) (f) \(f(x) = 2\) Equation Transcription: Text Transcription: f(x) = x^2 f(x) = x^3 f(x) = |x| f(x) = x + 1 f(x) = frac{x^5 -x}{1 + x^2} f(x) = 2

Suppose that the function \(f\) has domain all real numbers. Determine whether each function can be classified as even or odd. Explain. (a) \(g(x)=\frac{f(x)+f(-x)}{2}\) (b) \(h(x)=\frac{f(x)-f(-x)}{2}\) Equation Transcription: Text Transcription: f g(x) = frac{f(x) + f(-x)}{2} h(x) = frac{f(x) - f(-x)}{2}

Suppose that the function \(f\) has domain all real numbers. Show that \(f\) can be written as the sum of an even function and an odd function. [Hint: See Exercise 64] Equation Transcription: Text Transcription: f f

Use Theorem 0.2.3 to determine whether the graph has symmetries about the \(x\) - axis, the \(y\) - axis, or the origin. (a) \(x=5 y^{2}+9\) (b) \(x^{2}-2 y^{2}=3\) (c) \(x y=5\) Equation Transcription: Text Transcription: x y x = 5y^2 + 9 x^2 - 2y^2 = 3 xy = 5

Use Theorem 0.2.3 to determine whether the graph has symmetries about the \(x\) - axis, the \(y\) - axis, or the origin. (a) \(x^{4}=2 y^{3}+y\) (b) \(y=\frac{x}{3+x^{2}}\) (c) \(y^{2}=|x|-5\) Equation Transcription: Text Transcription: x y x^4 =2 y^3 + y y = frac{x}{3 + x^2} y^2 = | x| - 5

(i) Use a graphing utility to graph the equation in the first quadrant. [Note: To do this you will have to solve the equation for \(y\) in terms of \(x\).] (ii) Use symmetry to make a hand-drawn sketch of the entire graph. (iii) Confirm your work by generating the graph of the equation in the remaining three quadrants. \(9 x^{2}+4 y^{2}=36\) Equation Transcription: Text Transcription: y x 9x^2 + 4y^2 = 36

(i) Use a graphing utility to graph the equation in the first quadrant. [Note: To do this you will have to solve the equation for \(y\) in terms of \(x\).] (ii) Use symmetry to make a hand-drawn sketch of the entire graph. (iii) Confirm your work by generating the graph of the equation in the remaining three quadrants. \(4 x^{2}+16 y^{2}=16\) Equation Transcription: Text Transcription: y x 4x^2 + 16y^2 = 16

The graph of the equation \(x^{2 / 3}+y^{2 / 3}=1\), which is shown in the accompanying figure, is called a four-cusped hypocycloid. (a) Use Theorem 0.2.3 to confirm that this graph is symmetric about the \(x\) -axis, the \(y\) -axis, and the origin. (b) Find a function \(f\) whose graph in the first quadrant coincides with the four-cusped hypocycloid, and use a graphing utility to confirm your work. (c) Repeat part (b) for the remaining three quadrants. Equation Transcription: Text Transcription: x^? + y^? = 1 x y f

The equation \(y=|f(x)|\) can be written as \(y=\left\{\begin{array}{cc}f(x) & f(x) \geq 0 \\-f(x), & f(x)<0\end{array}\right.\) which shows that the graph of \(y=|f(x)|\) can be obtained from the graph of \(f(x)\) by retaining the portion that lies on or above the \(x\) -axis and reflecting about the \(x\) -axis the portion that lies below the \(x\) -axis. Use this method to obtain the graph of \(y = |2x ? 3|\) from the graph of \(y = 2x ? 3\). Equation Transcription: Text Transcription: y = |f(x)| y = {f(x) geq 0 -f(x), & f(x) < 0 y = |f(x)| y = f(x) x x x y = |2x - 3| y = 2x -3

Use the method described in Exercise 71. ? Sketch the graph of \(y=\left|1-x^{2}\right|\). Equation Transcription: Text Transcription: Y = |1 - x^2|

Use the method described in Exercise 71. Sketch the graph of (a) \(f(x) = | \cos x|\) (b) \(f(x) = \cos x + | \cos x|\). Equation Transcription: Text Transcription: f(x) = | cos x| f(x) = cos x + | cos x|

The greatest integer function, \(\lfloor x\rfloor\), is defined to be the greatest integer that is less than or equal to \(x\). For example, \(\lfloor 2.7\rfloor=2,[-2.3\rfloor=-3\), and \(\lfloor 4\rfloor=4\). In each part, sketch the graph of \(y=f(x)\). (a) \(f(x)=\lfloor x\rfloor\) (b) \(f(x)=\left\lfloor x^{2}\right\rfloor\) (c) \(f(x)=\lfloor x]^{2}\) (d) \(f(x)=\lfloor\sin x\rfloor\) Equation Transcription: ?? ?2.7? = 2, ?-2.3? = -3 ?4? = 4 ?? ?? ?? ?sin x? Text Transcription: floor x floor x y = f(x) floor 2.7 floor =2, floor - 2.3 floor = - 3 floor 4 floor = 4 f(x) = floor x floor f(x) = floor x^{2} floor f(x) = floor x floor^{2} f(x) = floor sin x floor

Is it ever true that \(f \circ g=g \circ f\) if \(f\) and g are non constant functions? If not, prove it; if so, give some examples for which it is true. Equation Transcription: Text Transcription: f circ g = g circ f f