a function $f$ is one-to-one if and only if every horizontal line intersects the graph of $f,$ at most, once

for a function $f,$ the inverse function $f^{-1}$ satisfies $f^{-1} (y) = x$ if $f (x) = y$

inverse trigonometric functions

the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

a function $f$ is one-to-one if $f (x_1) \ne f (x_2)$ if $x_1 \ne x_2$

a subset of the domain of a function $f$

Calculus Volume 1 · 1.4 Inverse Functions

Inverse Functions: practice problems

27 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 183
For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.
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Exercises 185
For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.
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Exercises 187
For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.
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Exercises 189
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
$f (x) = x^{2} - 4, x \geq 0$
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Exercises 190
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
$f (x) = 3 x - 4$
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Exercises 191
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
$f (x) = x^{3} + 1$
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Exercises 193
For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.
$f (x) = x - 1$
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Exercises 195
For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.
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Exercises 197
For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.
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Exercises 199
For the following exercises, use composition to determine which pairs of functions are inverses.
$f (x) = 8 x, g (x) = \frac{x}{8}$
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Exercises 201
For the following exercises, use composition to determine which pairs of functions are inverses.
$f (x) = 5 x - 7, g (x) = \frac{x + 5}{7}$
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Exercises 203
For the following exercises, use composition to determine which pairs of functions are inverses.
$f (x) = \frac{1}{x - 1}, x \neq = 1, g (x) = \frac{1}{x} + 1, x \neq = 0$
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Exercises 205
For the following exercises, use composition to determine which pairs of functions are inverses.
$f (x) = x^{2} + 2 x + 1, x \geq - 1, g (x) = - 1 + x, x \geq 0$
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Exercises 207
For the following exercises, evaluate the functions. Give the exact value.
$tan^{- 1} (\frac{3}{3})$
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Exercises 209
For the following exercises, evaluate the functions. Give the exact value.
$cot^{- 1} (1)$
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Exercises 211
For the following exercises, evaluate the functions. Give the exact value.
$cos^{- 1} (\frac{3}{2})$
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Exercises 213
For the following exercises, evaluate the functions. Give the exact value.
$sin (cos^{- 1} (\frac{2}{2}))$
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Exercises 215
For the following exercises, evaluate the functions. Give the exact value.
$tan^{- 1} (tan (- \frac{π}{6}))$
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Exercises 217
For the following exercises, evaluate the functions. Give the exact value.
[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function $V = f (x) = 500 (0.04 - x^{2})$ for $0 \leq x \leq 0.2.$
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Exercises 219
For the following exercises, evaluate the functions. Give the exact value.
[T] The cost to remove a toxin from a lake is modeled by the function
$C (p) = 75 p / (85 - p),$ where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.
Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.
Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.
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Exercises 221
For the following exercises, evaluate the functions. Give the exact value.
[T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $μ = 2 sin^{- 1} (\frac{1}{M}) .$
Find the Mach angle (to the nearest degree) for the following Mach numbers.
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Exercises 223
For the following exercises, evaluate the functions. Give the exact value.
[T] The average temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function
$T (x) = 5 + 18 sin [\frac{π}{6} (x - 4.6)],$
where $x$ is time in months and corresponds to January 1. Determine the day(s) (month and day) when the average temperature is Use the integer portion of your answer(s) as the month and calculate the day of the month from the decimal portion.
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Exercises 225
For the following exercises, evaluate the functions. Give the exact value.
[T] An object moving in simple harmonic motion is modeled by the function
$s (t) = - 6 cos (\frac{π t}{2}),$
where $s$ is measured in inches and $t$ is measured in seconds. Determine the first time when the distance moved is 4.5 in.
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Exercises 227
For the following exercises, evaluate the functions. Give the exact value.
[T] Use a calculator to evaluate $tan^{- 1} (tan (2.1))$ and $cos^{- 1} (cos (2.1)) .$ Explain the results of each.
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Review Exercises 313
True or False? Justify your answer with a proof or a counterexample.
A relation passing the horizontal line test is a function.
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Review Exercises 327
Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse $f^{- 1} (x)$ of the function. Justify your answer.
$f (x) = \frac{1}{x}$
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Review Exercises 329
For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.
$f (x) = x^{2} + 3 x + 4$
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Key terms · 5

horizontal line test: a function $f$ is one-to-one if and only if every horizontal line intersects the graph of $f,$ at most, once
inverse function: for a function $f,$ the inverse function $f^{-1}$ satisfies $f^{-1} (y) = x$ if $f (x) = y$
inverse trigonometric functions: the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions
one-to-one function: a function $f$ is one-to-one if $f (x_1) \ne f (x_2)$ if $x_1 \ne x_2$
restricted domain: a subset of the domain of a function $f$

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Inverse Functions: practice problems

27 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 183

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that decreases in a straight in until the origin, where it begins to increase in a straight line. The x intercept and y intercept are both at the origin.

Exercises 185

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a function that resembles a semi-circle, the top half of a circle. The function starts at the point (-3, 0) and increases until the point (0, 3), where it begins decreasing until it ends at the point (3, 0). The x intercepts are at (-3, 0) and (3, 0). The y intercept is at (0, 3).

Exercises 187

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of a curved function that is always increasing. The x intercept and y intercept are both at the origin.

Exercises 189

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) = x^{2} - 4, x \geq 0$

Exercises 190

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) = 3 x - 4$

Exercises 191

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

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

Exercises 193

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f (x) = x - 1$

Exercises 195

For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of an increasing straight line function labeled “f” that is always increasing. The x intercept is at (-2, 0) and y intercept are both at (0, 1).

Exercises 197

For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.

An image of a graph. The x axis runs from -8 to 8 and the y axis runs from -8 to 8. The graph is of an increasing straight line function labeled “f”. The function starts at the point (0, 1) and increases in straight line until the point (4, 6). After this point, the function continues to increase, but at a slower rate than before, as it approaches the point (8, 8). The function does not have an x intercept and the y intercept is (0, 1).

Exercises 199

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) = 8 x, g (x) = \frac{x}{8}$

Exercises 201

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) = 5 x - 7, g (x) = \frac{x + 5}{7}$

Exercises 203

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) = \frac{1}{x - 1}, x \neq = 1, g (x) = \frac{1}{x} + 1, x \neq = 0$

Exercises 205

For the following exercises, use composition to determine which pairs of functions are inverses.

$f (x) = x^{2} + 2 x + 1, x \geq - 1, g (x) = - 1 + x, x \geq 0$

Exercises 207