10.1 Finding Composite and Inverse Functions

Before you get started, take this readiness quiz.

If $f \left(x\right) = 2 x - 3$ and $g \left(x\right) = x^{2} + 2 x - 3 ,$ find $f \left(4\right) .$
If you missed this problem, review the reference.

Solve for $x ,$ $3 x + 2 y = 12 .$
If you missed this problem, review the reference.

Simplify: $5 \frac{\left(x + 4\right)}{5} - 4 .$
If you missed this problem, review the reference.

For functions $f \left(x\right) = 3 x - 2$ and $g \left(x\right) = 5 x + 1 ,$ find ⓐ $\left(f \circ g\right) \left(x\right)$ ⓑ $\left(g \circ f\right) \left(x\right)$ ⓒ $\left(f \cdot g\right) \left(x\right)$.

For functions $f \left(x\right) = 4 x - 3 ,$ and $g \left(x\right) = 6 x - 5 ,$ find ⓐ $\left(f \circ g\right) \left(x\right) ,$ ⓑ $\left(g \circ f\right) \left(x\right) ,$ and ⓒ $\left(f \cdot g\right) \left(x\right) .$

For functions $f \left(x\right) = x^{2} - 9 ,$ and $g \left(x\right) = 2 x + 5 ,$ find ⓐ $\left(f \circ g\right) \left(−2\right) ,$ ⓑ $\left(g \circ f\right) \left(−3\right) ,$ and ⓒ $\left(f \circ f\right) \left(4\right) .$

For functions $f \left(x\right) = x^{2} + 1 ,$ and $g \left(x\right) = 3 x - 5 ,$ find ⓐ $\left(f \circ g\right) \left(−1\right) ,$ ⓑ $\left(g \circ f\right) \left(2\right) ,$ and ⓒ $\left(f \circ f\right) \left(−1\right) .$

For each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.

ⓐ $\left\{\left(−3 , 27\right) , \left(−2 , 8\right) , \left(−1 , 1\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(2 , 8\right) , \left(3 , 27\right)\right\}$ and ⓑ $\left\{\left(0 , 0\right) , \left(1 , 1\right) , \left(4 , 2\right) , \left(9 , 3\right) , \left(16 , 4\right)\right\} .$

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

ⓐ $\left\{\left(−3 , −6\right) , \left(−2 , −4\right) , \left(−1 , −2\right) , \left(0 , 0\right) , \left(1 , 2\right) , \left(2 , 4\right) , \left(3 , 6\right)\right\}$ ⓑ $\left\{\left(−4 , 8\right) , \left(−2 , 4\right) , \left(−1 , 2\right) , \left(0 , 0\right) , \left(1 , 2\right) , \left(2 , 4\right) , \left(4 , 8\right)\right\}$

For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

ⓐ $\left\{\left(27 , −3\right) , \left(8 , −2\right) , \left(1 , −1\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(8 , 2\right) , \left(27 , 3\right)\right\}$ ⓑ $\left\{\left(7 , −3\right) , \left(−5 , −4\right) , \left(8 , 0\right) , \left(0 , 0\right) , \left(−6 , 4\right) , \left(−2 , 2\right) , \left(−1 , 3\right)\right\}$

Find the inverse of the function $\left\{\left(0 , 3\right) , \left(1 , 5\right) , \left(2 , 7\right) , \left(3 , 9\right)\right\} .$ Determine the domain and range of the inverse function.

Find the inverse of $\left\{\left(0 , 4\right) , \left(1 , 7\right) , \left(2 , 10\right) , \left(3 , 13\right)\right\} .$ Determine the domain and range of the inverse function.

Find the inverse of $\left\{\left(−1 , 4\right) , \left(−2 , 1\right) , \left(−3 , 0\right) , \left(−4 , 2\right)\right\} .$ Determine the domain and range of the inverse function.

Verify that the functions are inverse functions.

$f \left(x\right) = 4 x - 3$ and $g \left(x\right) = \frac{x + 3}{4} .$

Verify that the functions are inverse functions.

$f \left(x\right) = 2 x + 6$ and $g \left(x\right) = \frac{x - 6}{2} .$

Find the inverse of the function $f \left(x\right) = 5 x - 3 .$

Find the inverse of the function $f \left(x\right) = 8 x + 5 .$

Find the inverse of $f \left(x\right) = \sqrt[5]{2 x - 3} .$

Find the inverse of the function $f \left(x\right) = \sqrt[5]{3 x - 2} .$

Find the inverse of the function $f \left(x\right) = \sqrt[4]{6 x - 7} .$

Find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f \left(x\right) = 4 x + 3$ and $g \left(x\right) = 2 x + 5$

Find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f \left(x\right) = 6 x - 5$ and $g \left(x\right) = 4 x + 1$

Find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

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

Find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

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

Find the values described.

For functions $f \left(x\right) = 2 x^{2} + 3$ and $g \left(x\right) = 5 x - 1 ,$ find ⓐ $\left(f \circ g\right) \left(−2\right)$ ⓑ $\left(g \circ f\right) \left(−3\right)$ ⓒ $\left(f \circ f\right) \left(−1\right)$

Find the values described.

For functions $f \left(x\right) = 2 x^{3}$ and $g \left(x\right) = 3 x^{2} + 2 ,$ find ⓐ $\left(f \circ g\right) \left(−1\right)$ ⓑ $\left(g \circ f\right) \left(1\right)$ ⓒ $\left(g \circ g\right) \left(1\right)$

Determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

$\left\{\right. \left(−3 , 9\right) , \left(−2 , 4\right) , \left(−1 , 1\right) , \left(0 , 0\right)$,
$\left(1 , 1\right) , \left(2 , 4\right) , \left(3 , 9\right) \left.\right\}$

Determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

$\left\{\right. \left(−3 , −5\right) , \left(−2 , −3\right) , \left(−1 , −1\right)$,
$\left(0 , 1\right) , \left(1 , 3\right) , \left(2 , 5\right) , \left(3 , 7\right) \left.\right\}$

Find the inverse of each function. Determine the domain and range of the inverse function.

$\left\{\left(2 , 1\right) , \left(4 , 2\right) , \left(6 , 3\right) , \left(8 , 4\right)\right\}$

Find the inverse of each function. Determine the domain and range of the inverse function.

$\left\{\left(0 , −2\right) , \left(1 , 3\right) , \left(2 , 7\right) , \left(3 , 12\right)\right\}$

Find the inverse of each function. Determine the domain and range of the inverse function.

$\left\{\left(−2 , −3\right) , \left(−1 , −1\right) , \left(0 , 1\right) , \left(1 , 3\right)\right\}$

Determine whether or not the given functions are inverses.

$f \left(x\right) = x + 8$ and $g \left(x\right) = x - 8$

Determine whether or not the given functions are inverses.

$f \left(x\right) = 7 x$ and $g \left(x\right) = \frac{x}{7}$

Determine whether or not the given functions are inverses.

$f \left(x\right) = 7 x + 3$ and $g \left(x\right) = \frac{x - 3}{7}$

Determine whether or not the given functions are inverses.

$f \left(x\right) = \sqrt{x + 2}$ and $g \left(x\right) = x^{2} - 2 \left(x > 0\right)$

Find the inverse of each function.

$f \left(x\right) = x - 12$

Find the inverse of each function.

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

Find the inverse of each function.

$f \left(x\right) = \frac{x}{6}$

Find the inverse of each function.

$f \left(x\right) = 6 x - 7$

Find the inverse of each function.

$f \left(x\right) = −2 x + 5$

Find the inverse of each function.

$f \left(x\right) = x^{2} + 6 ,$ $x \geq 0$

Find the inverse of each function.

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

Find the inverse of each function.

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

Find the inverse of each function.

$f \left(x\right) = \sqrt{x - 2} ,$ $x \geq 2$

Find the inverse of each function.

$f \left(x\right) = \sqrt[3]{x - 3}$

Find the inverse of each function.

$f \left(x\right) = \sqrt[4]{9 x - 5} ,$ $x \geq \frac{5}{9}$

Find the inverse of each function.

$f \left(x\right) = \sqrt[5]{−3 x + 5}$

Explain how the graph of the inverse of a function is related to the graph of the function.