?(a) If \(f_0(x)=\frac{1}{2-x}\) and \(f_{n+1}=f_0 \circ f_n\) for \(n=0,1,2, \ldots\)

(a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?

The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.

(a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.

What is an increasing function?

What is a mathematical model?

Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f ) Rational function

Sketch by hand, on the same axes, the graphs of the following functions. (a) (b) (c) (d)

Draw, by hand, a rough sketch of the graph of each function. (a) (b) (c) (d) (e) (f) (g) (h)

Suppose that f has domain A and g has domain B. (a) What is the domain of f + g ? (b) What is the domain of fg ? (c) What is the domain of f/g ?

How is the composite function defined? What is its domain?

Suppose the graph of is given. Write an equation for each of the graphs that are obtained from the graph of as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis. (f ) Reflect about the y-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. ( i ) Stretch horizontally by a factor of 2. ( j ) Shrink horizontally by a factor of 2.

(a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? (b) If is a one-to-one function, how is its inverse function defined? How do you obtain the graph of from the graph of ?

(a) How is the inverse sine function \(f(x)=\sin ^{-1} x\) defined? What are its domain and range? (b) How is the inverse cosine function \(f(x)=\cos ^{-1} x\) defined? What are its domain and range? (c) How is the inverse tangent function \(f(x)=\tan ^{-1} x\) defined? What are its domain and range?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is any real number, then .

Use transformations to sketch the graph of the function.

Use transformations to sketch the graph of the function.

Use transformations to sketch the graph of the function.

Use transformations to sketch the graph of the function.

Determine whether is even, odd, or neither even nor odd. (a) (b) (c) (d) (e) (f)

Find an expression for the function whose graph consists of the line segment from the point s22, 2d to the point s21, 0d together with the top half of the circle with center the origin and radius 1.

If f(x) = ln x and \(g(x)=x^{2}-9\), find the functions (a) \(f \circ g\), (b) \(g \circ f\), (c) \(f \circ f\), (d) \(g \circ g\), and their domains.

Express the function as a composition of three functions.

Life expectancy has improved dramatically in recent decades. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2030.

A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the \(y\)-intercept of the graph and what does it represent? ________________ Equation Transcription: Text Transcription: y-intercept

If \(f(x)=2 x+4^{x}\), find \(f^{-1}(6)\). ________________ Equation Transcription: Text Transcription: f(x)=2x+4^x f^-1(6)

Find the inverse function of \(f(x)=\frac{2 x+3}{1-5 x}\).

Use the laws of logarithms to expand each expression. (a) \(\ln x \sqrt{x+1}\) (b) \(\log _{2} \sqrt{\frac{x^{2}+1}{x-1}}\) Equation Transcription: Text Transcription: ln x sqrt x+1 log_2 sqrt x^+1/x-1

Express as a single logarithm. (a) \(\frac{1}{2} \ln x-2 \ln \left(x^{2}+1\right)\) (b) \(\ln (x-3)+\ln (x+3)-2 \ln \left(x^{2}-9\right)\) Equation Transcription: Text Transcription: 1/2ln x-2 ln(x^2+1) ln(x-3)+ ln(x+3)-2 ln(x^2-9)

Find the exact value of each expression. (a) \(e^{2 \ln 5}\) (b) \(\log _{6} 4+\log _{6} 54\) (c) \(\tan \left(\arcsin \frac{4}{5}\right)\) Equation Transcription: Text Transcription: e^2ln5 log_6 4+log_6 54 tan(arcsin 4/5)

Find the exact value of each expression. (a) \(\ln \frac{1}{e^{3}}\) (b) \(\sin \left(\tan ^{-1} 1\right)\) (c) \(10^{-3 \log 4}\) Equation Transcription: Text Transcription: ln the fraction with numerator of 1 and with denominator 3 cube sin inverse tangent of 1 10 raise to -3 log to the 4th power

Solve the equation for . Give both an exact value and a decimal approximation, correct to three decimal places.

Solve the equation for \(x\) Give both an exact value and a decimal approximation, correct to three decimal places. \(\ln x^{2}=5\) Equation Transcription: Text Transcription: x ln x^2 = 5

Solve the equation for x. Give both an exact value and a decimal approximation, correct to three decimal places. \(e^{e^{x}}=10\)

Solve the equation for \(x\). Give both an exact value and a decimal approximation, correct to three decimal places. \(\cos ^{-1} x=2\) Equation Transcription: ???? Text Transcription: x cos^-1 x=2

Solve the equation for \(x\). Give both an exact value and a decimal approximation, correct to three decimal places. \(\tan ^{-1}\left(3 x^{2}\right)=\frac{\pi}{4}\)

Solve the equation for x. Give both an exact value and a decimal approximation, correct to three decimal places. ln x - 1 = ln(5 + x) - 4

The viral load for an HIV patient is 52.0 RNA copies mL before treatment begins. Eight days later the viral load is half of the initial amount. (a) Find the viral load after 24 days. (b) Find the viral load \(V(t)\) that remains after \(t\) days. (c) Find a formula for the inverse of the function \(V\) and explain its meaning. (d) After how many days will the viral load be reduced to 2.0 RNA copies/mL? ________________ Equation Transcription: Text Transcription: V(t) t V

The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is \(P(t)=\frac{100,000}{100+900 e^{-t}}\) where \(t\) is measured in years. (a) Graph this function and estimate how long it takes for the population to reach 900 . (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900 . Compare with the result of part (a). ________________ Equation Transcription: Text Transcription: P(t)=100,000/100+900e^-t t

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is a function, then f(s+t)=f(s)+f(t).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(s)=f(t)\), then \(=s=t\). Equation Transcription: Text Transcription: f(s) = f(t) s = t

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is a function, then . Equation Transcription: Text Transcription: f f(3x) = 3f(x).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the function \(f\) has an inverse and \(f(2)=3\), then \(f^{-1}(3)=2\). Equation Transcription: Text Transcription: f f(2) = 3 f-1(3) = 2

What is a mathematical model?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are functions, then \(f \circ g=g \circ f\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is one-to-one, then . Equation Transcription: Text Transcription: f f-1(x) = 1f(x).

Draw, by hand, a rough sketch of the graph of each function. (a) \(y=\sin x\) (b) \(y=\tan x\) (c) \(y=e^{x}\) (d) \(y=\ln x\) (e) \(y=1 / x\) (f) \(y=|x|\) (g) \(y=\sqrt{x}\) (h) \(y=\tan ^{-1} x\) Equation Transcription: Text Transcription: y=sin x y=tan x y=e^x y=ln x y=1/x y=|x| y= sqrt x y=tan^-1 x

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If If 0<a<b, then ln a<ln b

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If x > 0, then \((\ln x)^{6}=6 \ln x\).

Suppose the graph of \(f\) is given. Write an equation for each of the graphs that are obtained from the graph of \(f\) as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the \(x\)-axis. (f ) Reflect about the \(y\)-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. ( i ) Stretch horizontally by a factor of 2. ( j ) Shrink horizontally by a factor of 2. Equation Transcription: Text Transcription: f x y

(a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? (b) If \(f\) is a one-to-one function, how is its inverse function \(f^{-1}\) defined? How do you obtain the graph of \(f^{-1}\) from the graph of \(f\)? Equation Transcription: Text Transcription: f f^-1

(a) How is the inverse sine function \(f(x)=\sin ^{-1} x\) defined? What are its domain and range? (b) How is the inverse cosine function \(f(x)=\cos ^{-1} x\) defined? What are its domain and range? (c) How is the inverse tangent function \(f(x)=\tan ^{-1} x\) defined? What are its domain and range? Equation Transcription: Text Transcription: f(x) = sin^-1x f(x) = cos^-1x f(x) = tan^-1x

Express as a single logarithm. (a) \(\frac{1}{2} \ln x-2 \ln \left(x^{2}+1\right)\) (b) \(\ln (x-3)+\ln (x+3)-2 \ln \left(x^{2}-9\right)\) Equation Transcription: Text Transcription: 1/2ln x-2 ln(x^2+1) ln(x-3)+ ln(x+3)-2 ln(x^2-9)

Find the exact value of each expression. (a) \(e^{2 \ln 5}\) (b) \(\log _{6} 4+\log _{6} 54\) (c) \(\tan \left(\arcsin \frac{4}{5}\right)\) Equation Transcription: Text Transcription: e^2ln5 log_6 4+log_6 54 tan(arcsin 4/5)

Find the exact value of each expression. (a) \(\ln \frac{1}{e^{3}}\) (b) \(\sin \left(\tan ^{-1} 1\right)\) (c) \(10^{-3 \log 4}\) Equation Transcription: Text Transcription: ln1/e^3 sin(tan^-1 1) 10^3log4

Solve the equation for \(x\). Give both an exact value and a decimal approximation, correct to three decimal places. \(e^{2 x}=3\) ________________ Equation Transcription: Text Transcription: x e^2x=3

Solve the equation for \(x\) Give both an exact value and a decimal approximation, correct to three decimal places. \(\ln x^{2}=5\) Equation Transcription: Text Transcription: x ln x^2 = 5

Solve the equation for . Give both an exact value and a decimal approximation, correct to three decimal places.

Solve the equation for \(x\). Give both an exact value and a decimal approximation, correct to three decimal places. \(\cos ^{-1} x=2\) Equation Transcription: ???? Text Transcription: x cos^-1 x=2

Solve the equation for x. Give both an exact value and a decimal approximation, correct to three decimal places. \(\tan ^{-1}\left(3 x^{2}\right)=\frac{\pi}{4}\)

Solve the equation for \(x\). Give both an exact value and a decimal approximation, correct to three decimal places. \(\ln x-1=\ln (5+x)-4\) Equation Transcription: Text Transcription: x ln x-1=ln(5+x)-4

The viral load for an HIV patient is 52.0 RNA copies mL before treatment begins. Eight days later the viral load is half of the initial amount. (a) Find the viral load after 24 days. (b) Find the viral load V(t) that remains after t days. (c) Find a formula for the inverse of the function V and explain its meaning. (d) After how many days will the viral load be reduced to 2.0 RNA copies/mL?

The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is where is measured in years. (a) Graph this function and estimate how long it takes for the population to reach 900 . (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900 . Compare with the result of part (a).

One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.

The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.

Solve the equation .

Solve the inequality \(|x-1|-|x-3| \geqslant 5\).

Sketch the graph of the function

Sketch the graph of the function \(g(x)=\left|x^{2}-1\right|-\left|x^{2}-4\right|\).

Draw the graph of the equation

Sketch the region in the plane consisting of all points such that

The notation max {a, b, …} means the largest of the numbers a, b, …. Sketch the graph of each function. (a) f(x) = max {x, 1/x} (b) f(x) = max {sin x, cos x} (c) \(f(x)=\max \left\{x^{2}, 2+x, 2-x\right\}\)

Sketch the region in the plane defined by each of the following equations or inequalities. (a) max {x, 2y} = 1 (b) \(-1 \leqslant \max \{x, 2 y\} \leqslant 1\) (c) \(\max \left\{x, y^{2}\right\}=1\)

Show that if \(x>0\) and \(x \neq 1\), then \(\frac{1}{\log _{2} x}+\frac{1}{\log _{3} x}+\frac{1}{\log _{5} x}=\frac{1}{\log _{x} x}\) Equation Transcription: Text Transcription: x>0 x \neq 1 1/log_2 ?x+1/log_3 ?x+1/log_5 ?x=1/log_x ?x

Find the number of solutions of the equation \(\sin x=\frac{x}{100}\) Equation Transcription: Text Transcription: sin? x=x/100

Find the exact value of \(\sin \frac{\pi}{100}+\sin \frac{2 \pi}{100}+\sin \frac{3 \pi}{100}+\cdots+\sin \frac{200 \pi}{100}\) Equation Transcription: Text Transcription: Sin pi/?100+sin ?2pi/100+sin ?3pi/100+...+sin? 200pi/100

(a) Show that the function \(f(x)=\ln \left(x+\sqrt{x^2+1}\right)\) is an odd function. (b) Find the inverse function of f. Equation Transcription: Text Transcription: f f(x)=ln? (x+x2+1)

Solve the inequality \(\ln \left(x^{2}-2 x-2\right) \leqslant 0\) . Equation Transcription: ?0 Text Transcription: ln? (x2?2x?2) \leqslant 0

Use indirect reasoning to prove that \(\log_{2} 5\) is an irrational number.

A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of \(30 \mathrm{mi} / \mathrm{h}\); she drives the second half at \(60 \mathrm{mi} / \mathrm{h}\). What is her average speed on this trip? Equation Transcription: Text Transcription: 30 mi/h 60 mi/h

Is it true that \(f \circ(g+h)=f \circ g+f \circ h\)

Prove that if \(n\)is a positive integer, then \(7^{n}-1\) is divisible by 6 . Equation Transcription: Text Transcription: n 7^n-1

Prove that \(1+3+5+\cdots+(2 n-1)=n^{2}\). Equation Transcription: Text Transcription: 1+3+5+cdots+(2n-1)=n^2

If \(f_{0}(x)=x^{2}\) and \(f_{n+1}(x)=f_{0}\left(f_{n}(x)\right)\) for \(n=0,1,2, \ldots\), find a formula for \(f_{n}(x)\). Equation Transcription: Text Transcription: f_0(x)=x^2 f_n+1(x)=f_0(f_n(x)) n=0,1,2,... f_n(x)

(a) If \(f_0(x)=\frac{1}{2-x}\) and \(f_{n+1}=f_0 \circ f_n\) for \(n=0,1,2, \ldots\), find an expression for \(f_n(x)\) and use mathematical induction to prove it. (b) Graph \(f_0, f_1, f_2, f_3\) on the same screen and describe the effects of repeated composition.