3.7 Inverse Functions

For functions $f \left(x\right) = 2 x - 7$ , $g \left(x\right) = \frac{x + 7}{2}$ , find:

1. ⓐ $g \left(5\right)$
2. ⓑ $f \left(g \left(5\right)\right)$
3. ⓒ $f \left(g \left(x\right)\right)$

If for a particular one-to-one function $f \left(2\right) = 4$ and $f \left(5\right) = 12 ,$ what are the corresponding input and output values for the inverse function?

Given that $h^{- 1} \left(6\right) = 2 ,$ what are the corresponding input and output values of the original function $h ?$

If $f \left(x\right) = \frac{1}{x + 2}$ and $g \left(x\right) = \frac{1}{x} - 2 ,$ is $g = f^{- 1} ?$

If $f \left(x\right) = x^{3} - 4$ and $g \left(x\right) = \sqrt[ 3]{x + 4} ,$ is $g = f^{- 1} ?$

If $f \left(x\right) = x^{3}$ (the cube function) and $g \left(x\right) = \frac{1}{3} x ,$ is $g = f^{- 1} ?$

If $f \left(x\right) = \left(x - 1\right)^{3} \text{and} g \left(x\right) = \sqrt[3]{x} + 1 ,$ is $g = f^{- 1} ?$

Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in the table. We restrict the domain in such a fashion that the function assumes all y-values exactly once.

The domain of function $f$ is $\left(1 , \infty\right)$ and the range of function $f$ is $\left(−∞ , −2\right) .$ Find the domain and range of the inverse function.

A function $f \left(t\right)$ is given in the table, showing distance in miles that a car has traveled in $t$ minutes. Find and interpret $f^{- 1} \left(70\right) .$

Using the table, find and interpret ⓐ $f \left(60\right) ,$ and ⓑ $f^{- 1} \left(60\right) .$

Using the graph in the figure, ⓐ find $g^{- 1} \left(1\right) ,$ and ⓑ estimate $g^{- 1} \left(4\right) .$

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

Solve for $x$ in terms of $y$ given $y = \frac{1}{3} \left(x - 5\right) .$

Find the inverse of the function $f \left(x\right) = \frac{2}{x - 3} + 4.$

Find the inverse of the function $f \left(x\right) = 2 + \sqrt{x - 4} .$

What is the inverse of the function $f \left(x\right) = 2 - \sqrt{x} ?$ State the domains of both the function and the inverse function.

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Can a function be its own inverse? Explain.

How do you find the inverse of a function algebraically?

Find $f^{- 1} \left(x\right)$ for each function.

$f \left(x\right) = x + 3$

Find $f^{- 1} \left(x\right)$ for each function.

$f \left(x\right) = 2 - x$

Find $f^{- 1} \left(x\right)$ for each function.

$f \left(x\right) = \frac{x}{x + 2}$

Find a domain on which each function $f$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of $f$ restricted to that domain.

$f \left(x\right) = \left(x + 7\right)^{2}$

Find a domain on which each function $f$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of $f$ restricted to that domain.

$f \left(x\right) = x^{2} - 5$

Find a domain on which each function $f$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of $f$ restricted to that domain.

Given $f \left(x\right) = \frac{x}{2 + x}$ and $g \left(x\right) = \frac{2 x}{1 - x} :$

1. ⓐFind $f \left(g \left(x\right)\right)$ and $g \left(f \left(x\right)\right) .$
2. ⓑWhat does the answer tell us about the relationship between $f \left(x\right)$ and $g \left(x\right) ?$

Use function composition to verify that $f \left(x\right)$ and $g \left(x\right)$ are inverse functions.

$f \left(x\right) = \sqrt[3]{x - 1}$ and $g \left(x\right) = x^{3} + 1$

Use a graphing utility to determine whether each function is one-to-one.

$f \left(x\right) = \sqrt{x}$

Use a graphing utility to determine whether each function is one-to-one.

$f \left(x\right) = −5 x + 1$

Find $f \left(0\right) .$

Find $f^{- 1} \left(0\right) .$

If the complete graph of $f$ is shown, find the domain of $f .$

Evaluate or solve, assuming that the function $f$ is one-to-one.

If $f \left(6\right) = 7 ,$ find $f^{- 1} \left(7\right) .$

Evaluate or solve, assuming that the function $f$ is one-to-one.

If $f^{- 1} \left(- 4\right) = - 8 ,$ find $f \left(- 8\right) .$

Find $f \left(1\right) .$

Find $f^{- 1} \left(0\right) .$

Use the tabular representation of $f$ in the table to create a table for $f^{- 1} \left(x\right) .$

To convert from $x$ degrees Celsius to $y$ degrees Fahrenheit, we use the formula $f \left(x\right) = \frac{9}{5} x + 32.$ Find the inverse function, if it exists, and explain its meaning.

A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, $t ,$ in hours given by $d \left(t\right) = 50 t .$ Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function $t \left(d\right) .$ Find $t \left(180\right)$ and interpret its meaning.

Determine whether the relation is a function.

$\left\{\left(a , b\right) , \left(c , d\right) , \left(e , d\right)\right\}$

Determine whether the relation is a function.

$y^{2} + 4 = x ,$ for $x$ the independent variable and $y$ the dependent variable

Evaluate $f \left(- 3\right) ; f \left(2\right) ; f \left(- a\right) ; - f \left(a\right) ; f \left(a + h\right) .$

$f \left(x\right) = - 2 x^{2} + 3 x$

Determine whether the functions are one-to-one.

$f \left(x\right) = - 3 x + 5$

$f \left(−2\right)$

If $f \left(x\right) = 1 ,$ then solve for $x .$

Use the function $h \left(t\right) = - 16 t^{2} + 80 t$ to find the values in simplest form.

$\frac{h \left(a\right) - h \left(1\right)}{a - 1}$

Find the domain of each function, expressing answers using interval notation.

$f \left(x\right) = \frac{x - 3}{x^{2} - 4 x - 12}$

Find the average rate of change of the functions from $x = 1 \text{to}\textrm{ } x = 2.$

$f \left(x\right) = 10 x^{2} + x$

Use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Find the local extrema for the function graphed in the exercise.

Find $\left(f \circ g\right) \left(x\right)$ and $\left(g \circ f\right) \left(x\right)$ for each pair of functions.

$f \left(x\right) = 3 x + 2 , g \left(x\right) = 5 - 6 x$

Find $\left(f \circ g\right) \left(x\right)$ and $\left(g \circ f\right) \left(x\right)$ for each pair of functions.

$f \left(x\right) = \sqrt{x + 2} , g \left(x\right) = \frac{1}{x}$

Find $\left(f \circ g\right)$ and the domain for $\left(f \circ g\right) \left(x\right)$ for each pair of functions.

$f \left(x\right) = \frac{x + 1}{x + 4} , g \left(x\right) = \frac{1}{x}$

Find $\left(f \circ g\right)$ and the domain for $\left(f \circ g\right) \left(x\right)$ for each pair of functions.

$f \left(x\right) = \frac{1}{x} , g \left(x\right) = \sqrt{x}$

Express each function $H$ as a composition of two functions $f$ and $g$ where $H \left(x\right) = \left(f \circ g\right) \left(x\right) .$

$H \left(x\right) = \sqrt{\frac{2 x - 1}{3 x + 4}}$

Determine whether each function below is even, odd, or neither.

$f \left(x\right) = 3 x^{4}$

Determine whether each function below is even, odd, or neither.

$h \left(x\right) = \frac{1}{x} + 3 x$

Find $f^{- 1} \left(x\right)$ for each function.

$f \left(x\right) = 9 + 10 x$

Find a domain on which the function $f$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of $f$ restricted to that domain.

$f \left(x\right) = x^{2} + 1$

Use a graphing utility to determine whether each function is one-to-one.

If $f \left(5\right) = 2 ,$ find $f^{- 1} \left(2\right) .$

Determine whether each of the following relations is a function.

$y = 2 x + 8$

Evaluate the function $f \left(x\right) = - 3 x^{2} + 2 x$ at the given input.

$f \left(−2\right)$

Evaluate the function $f \left(x\right) = - 3 x^{2} + 2 x$ at the given input.

Show that the function $f \left(x\right) = - 2 \left(x - 1\right)^{2} + 3$ is not one-to-one.

Evaluate the function $f \left(x\right) = - 3 x^{2} + 2 x$ at the given input.

Given $f \left(x\right) = 2 x^{2} - 5 x ,$ find $f \left(a + 1\right) - f \left(1\right)$ in simplest form.

Evaluate the function $f \left(x\right) = - 3 x^{2} + 2 x$ at the given input.

Find the average rate of change of the function $f \left(x\right) = 3 - 2 x^{2} + x$ by finding $\frac{f \left(b\right) - f \left(a\right)}{b - a}$ in simplest form.

Use the functions $f \left(x\right) = 3 - 2 x^{2} + x \text{and}\textrm{ } g \left(x\right) = \sqrt{x}$ to find the composite functions.

$\left(g \circ f\right) \left(1\right)$

Determine whether the functions are even, odd, or neither.

$f \left(x\right) = - \frac{5}{x^{2}} + 9 x^{6}$

Determine whether the functions are even, odd, or neither.

$f \left(x\right) = \frac{1}{x}$

Find the inverse of the function.

$f \left(x\right) = 3 x - 5$

On what intervals is the function increasing?

Approximate the local minimum of the function. Express the answer as an ordered pair. The ordered pair should include both the $x$ value as well as the $g \left(x\right)$ value.

Find $f \left(2\right) .$

Write an equation for the piecewise function.

Solve the equation $f \left(x\right) = 5.$

Is the function represented by the graph one-to-one?

Given $f \left(x\right) = - 2 x + 11 ,$ find $f^{- 1} \left(x\right) .$

Solve the absolute value equation.

$\left|x + 4\right| = 18$

Solve the inequality and express the solution using interval notation.

$\left|3 x - 2\right| < 7$

Determine whether the functions are even, odd, or neither.

Solve $\left|2 x - 3\right| = 17.$