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For b > 0, what are the domain and range of f(x) = b^{x}?

Give an example of a function that is one-to-one on the entire real number line.

Explain why a function that is not one-to-one on an interval I cannot have an inverse function on I.

Explain with pictures why (a, b) is on the graph of f whenever (b, a) is on the graph of f^{-1}.

Sketch a function that is one-to-one and positive for x ≥ 0. Make a rough sketch of its inverse.

Express the inverse of f(x) = 3x - 4 in the form y = f^{-1}(x).

Explain the meaning of log_{b} x.

How is the property b^{x + y} = b^{x}b^{y} related to the property log_{b} (xy) = log_{b} x + log_{b} y?

For b > 0 with b ≠ 1, what are the domain and range of f(x) = log_{b}x and why?

Express 2^{5} using base e.

Find three intervals on which f is one-to-one, making each interval as large as possible.

Find four intervals on which f is one-to-one, making each interval as large as possible.

Sketch a graph of a function that is one-to-one on the interval (- ∞, 0] but is not one-to-one on (- ∞, ∞).

Sketch a graph of a function that is one-to-one on the intervals (- ∞, -2], and [-2, ∞) but is not one-to-one on (- ∞, ∞).

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

f(x) = 3x + 4

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

f(x) = |2x + 1|

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

f(x) = 1/(x - 5)

f(x) = -(6 - x)^{2}

f(x) = 1/x^{2}

f(x) = x^{2} - 2x + 8

a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 2x

a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 1/4 x + 1

a. Find the inverse of each function (on the given interval, if specified) and write it in the form y = f^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 6 - 4x

^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 3x^{3}

^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 3x + 5

^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = x^{2} + 4, for x ≥ 0

^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = √x + 2, for x ≥ -2

^{-1}(x).

b. Verify the relationships f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

f(x) = 2/(x^{2} + 1), for x ≥ 0

The unit circle x^{2} + y^{2} = 1 consists of four one-to-one functions, f_{1}(x), f_{2}(x), f_{3}(x), and f_{4}(x) (see figure).

a. Find the domain and a formula for each function.

b. Find the inverse of each function and write it as y = f^{-1}(x).

The equation y^{4} = 4x^{2} is associated with four one-to-one functions f_{1}(x), f_{2}(x), f_{3}(x), and f_{4}(x) (see figure).

a. Find the domain and a formula for each function.

b. Find the inverse of each function and write it as y = f^{-1}(x).

Find the inverse function (on the given interval, if specified) and graph both f and f^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = 8 - 4x

Find the inverse function (on the given interval, if specified) and graph both f and f^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = 4x - 12

Find the inverse function (on the given interval, if specified) and graph both f and f^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = √x, for x ≥ 0

^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = √3 - x, for x ≤ 3

^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = x^{4} + 4, for x ≥ 0

^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = 6/(x^{2} - 9), for x > 3

^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = x^{2} - 2x + 6, for x ≥ 1

^{-1} on the same set of axes. Check your work by looking for the required symmetry in the graphs.

f(x) = -x^{2} - 4x - 3, for x ≤ -2

Sketch the graph of the inverse function.

Sketch the graph of the inverse function.

Solve the following equations.

log_{10} x = 3

Solve the following equations.

log_{5} x = -1

Solve the following equations.

log_{8} x = 1/3

Solve the following equations.

log_{b} 125 = 3

Solve the following equations.

ln x = -1

Solve the following equations.

ln y = 3

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} x/y

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} x^{2}

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} xz

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} √xy/z

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} √x/^{3}√z

Assume log_{b} x = 0.36, log_{b} y = 0.56, and log_{b} z = 0.83. Evaluate the following expressions.

log_{b} b^{2}x^{5/2}/√y

Solve the following equations.

7^{x} = 21

Solve the following equations.

2^{x} = 55

Solve the following equations.

3^{3x - 4} = 15

Solve the following equations.

5^{3x} = 29

One hundred grams of a particular radioactive substance decays according to the function m(t) = 100 e^{-t/650}, where t > 0 measures time in years. When does the mass reach 50 grams?

The population P of a small town grows according to the function P(t) = 100 e^{t/50}, where t measures the number of years after 2010. How long does it take the population to double?

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

log_{2} 15

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

log_{3} 30

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.

log_{4} 40

log_{6} 60

Convert the following expressions to the indicated base.

2^{x} using base e

Convert the following expressions to the indicated base.

3^{sin x} using base e

Convert the following expressions to the indicated base.

ln |x| using base 5

Convert the following expressions to the indicated base.

log_{2} (x^{2} + 1) using base e

Convert the following expressions to the indicated base.

a^{1/ln a} using base e, for a > 0 and a ≠ 1

Convert the following expressions to the indicated base.

a^{1/log10 a} using base 10, for a > 0 and a ≠ 1

Determine whether the following statements are true and give an explanation or counterexample.

a. If y = 3^{x}, then x = ^{3}√y.

b. log_{b} x/log_{b} y = log_{b} x - log_{b} y

c. log_{5} 4^{6} = 4 log_{5} 6

d. 2 = 10^{log10 2}

e. 2 = ln 2^{e}

f. If f(x) = x^{2} + 1, then f^{-1}(x) = 1/(x^{2} + 1).

g. If f(x) = 1/x, then f^{-1}(x) = 1/x.

The following figure shows the graphs of y = 2^{x}, y = 3^{x}, y = 2^{-x}, and y = 3^{-x}. Match each curve with the correct function.

The following figure shows the graphs of y = log_{2} x, y = log_{4} x, and y = log_{10} x. Match each curve with the correct function.

Without using a graphing utility, sketch the graph of y = 2^{x}. Then on the same set of axes, sketch the graphs of y = 2^{-x}, y = 2^{x - 1}, y = 2^{x} + 1, and y = 2^{2x}.

Without using a graphing utility, sketch the graph of y = log_{2} x. Then on the same set of axes, sketch the graphs of y = log_{2}(x - 1), y = log_{2} x^{2}, y = (log_{2} x)^{2}, and y = log_{2} x + 1.

Use any means to approximate the intersection point(s) of the graphs of f(x) = e^{x} and g(x) = x^{123}.

Find all the inverses associated with the following functions and state their domains.

f(x) = (x + 1)^{3}

Find all the inverses associated with the following functions and state their domains.

f(x) = (x - 4)^{2}

Find all the inverses associated with the following functions and state their domains.

f(x) = 2/(x^{2} + 2)

Find all the inverses associated with the following functions and state their domains.

f(x) = 2x/(x + 2)

A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function p(t) = 150 ∙ 2^{t/12}, where t is the number of hours after the first observation.

a. Verify that p(0) = 150, as claimed.

b. Show that the population doubles every 12 hr, as claimed.

c. What is the population 4 days after the first observation?

d. How long does it take the population to triple in size?

e. How long does it take the population to reach 10,000?

A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function Q(t) = a(1 - e^{-t/c}), where t is measured in seconds, and a and c > 0 are physical constants. The steady-state charge is the value that Q(t) approaches as t becomes large.

a. Graph the charge function for t ≥ 0 using a = 1 and c = 10. Find a graphing window that shows the full range of the function.

b. Vary the value of a while holding c fixed. Describe the effect on the curve. How does the steady-state charge vary with a?

c. Vary the value of c while holding a fixed. Describe the effect on the curve. How does the steady-state charge vary with c?

d. Find a formula that gives the steady-state charge in terms of a and c.

The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h = f(t) = 64t - 16t^{2}, where t is measured in seconds after the hit.

a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?

b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = f^{-1}(h).

c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = f^{-1}(h).

d. At what time is the ball at a height of 30 ft on the way up?

e. At what time is the ball at a height of 10 ft on the way down?

The velocity of a skydiver (in m/s) t seconds after jumping from a plane is v(t) = 600(1 - e^{-kt/60})/k, where k > 0 is a constant. The terminal velocity of the skydiver is the value that v(t) approaches as t becomes large. Graph v with k = 11 and estimate the terminal velocity.

Assume that b > 0 and b ≠ 1. Show that log_{1/b} x = - log_{b} x.

Use the following steps to prove that log_{b} xy = log_{b} x + log_{b} y.

a. Let x = b_{p} and y = b^{q}. Solve these expressions for p and q, respectively.

b. Use property E1 for exponents to express xy in terms of b, p, and q.

c. Compute log_{b} xy and simplify.

Modify the proof outlined in Exercise 84 and use property E2 for exponents to prove that log_{b} (x/y) = log_{b} x - log_{b} y.

Use the following steps to prove that log_{b} x^{z} = z log_{b} x.

a. Let x = b^{p}. Solve this expression for p.

b. Use property E3 for exponents to express x^{z} in terms of b and p.

c. Compute log_{b} x^{z} and simplify.

Consider the quartic polynomial y = f(x) = x^{4} - x^{2}.

a. Graph f and estimate the largest intervals on which it is one-to- one. The goal is to find the inverse function on each of these intervals.

b. Make the substitution u = x^{2} to solve the equation y = f(x) for x in terms of y. Be sure you have included all possible solutions.

c. Write each inverse function in the form y = f^{-1}(x) for each of the intervals found in part (a).

a. Let g(x) = 2x + 3 and h(x) = x^{3}. Consider the composite function f(x) = g(h(x)). Find f^{-1} directly and then express it in terms of g^{-1} and h^{-1}.

b. Let g(x) = x^{2} + 1 and h(x) = √x. Consider the composite function f(x) = g(h(x)). Find f^{-1} directly and then express it in terms of g^{-1} and h^{-1}.

c. Explain why if g and h are one-to-one, the inverse of f(x) = g(h(x)) exists.

Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f(x) = x^{3} + ax. Find the inverse of the following cubics using the substitution (known as Vieta’s substitution) x = z - a/(3z). Be sure to determine where the function is one-to-one.

f(x) = x^{3} + 2x

Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic y = f(x) = x^{3} + ax. Find the inverse of the following cubics using the substitution (known as Vieta’s substitution) x = z - a/(3z). Be sure to determine where the function is one-to-one.

f(x) = x^{3} + 4x - 1

Prove that (log_{b} c) (log_{c} b) = 1, for b > 0, c > 0, b ≠ 1, and c ≠ 1.

Define the six trigonometric functions in terms of the sides of a right triangle.

Explain how a point P(x, y) on a circle of radius r determines an angle θ and the values of the six trigonometric functions at θ.

How is the radian measure of an angle determined?

Explain what is meant by the period of a trigonometric function. What are the periods of the six trigonometric functions?

What are the three Pythagorean identities for the trigonometric functions?

How are the sine and cosine functions related to the other four trigonometric functions?

Where is the tangent function undefined?

What is the domain of the secant function?

Explain why the domain of the sine function must be restricted in order to define its inverse function.

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