1.4: Composition and Inverses

Composition of Functions

Example $\PageIndex{1}$

Sociologists in Holland determine that the number of people $y$ waiting in a water ride at an amusement park is given by

$$y = \dfrac{1}{50}C^2 + C + 2 \nonumber$$

where $C$ is the temperature in degrees $C$. The formula to convert Fahrenheit to Celsius $C$ is given by

$$C = \dfrac{5}{9}F + \dfrac{160}{9}. \nonumber$$

To get a function of $F$ we compose the two function:

$$y(C(F)) = \left(\dfrac{1}{50}\right)\left[\dfrac{5}{9}F + \dfrac{160}{9}\right]^2 + \left[\dfrac{5}{9}F + \dfrac{160}{9}\right] + 2 \nonumber$$

Exercise $\PageIndex{1}$

If

• $f(x) = 3x + 2$
• $g(x) = 2x^2 + 1$
• $h(x) = \sqrt{x-2}$
• $c(x) = 4$

Find

1. $f(g(x))$
2. $f(h(x))$
3. $f(f(x))$
4. $h(c(x))$
5. $c(f(g(h(x))))$

1-1 Functions

Definition: 1-1 (one-to-one)

A function $f(x)$ is 1-1 if

$$f(a) = f(b)$$

implies that

$$a = b.$$

Example $\PageIndex{2}$

If

$$f(x) = 3x + 1 \nonumber$$

then

$$3a + 1 = 3b + 1 \nonumber$$

implies that

$$3a = 3b \nonumber$$

hence

$$a = b \nonumber$$

therefore $f(x)$ is 1-1.

Example $\PageIndex{3}$

If

$$f(x) = x^2 \nonumber$$

then

$$a^2 = b^2 \nonumber$$

implies that

$$a^2-b^2 = 0 \nonumber$$

or that

$$(a - b)(a + b) = 0 \nonumber$$

hence

$$a = b \text{ or } a = -b \nonumber$$

For example

$$f (2) = f (-2) = 4 \nonumber$$

Hence $f(x)$ is not 1-1.

Horizontal Line Test

If every horizontal line passes through $f(x)$ at most once then $f(x)$ is 1-1.

Inverse Functions

Definition: Inverse function

A function $g(x)$ is an inverse of $f(x)$ if

$$f(g(x)) = g(f(x)) = x.$$

Example $\PageIndex{4}$

The volume of a lake is modeled by the equation

$$V(t) = \dfrac{1}{125}h^3. \nonumber$$

Show that the inverse is

$$h(N) = 5V^{\frac{1}{3}}. \nonumber$$

Solution: We have

$$h(V(h)) = 5(\dfrac{1}{125}h^3)^{\frac{1}{3}} = \dfrac{5}{5}h = h \nonumber$$

and

$$v(h(V)) = \dfrac{1}{125}(5V^{\frac{1}{3}})^3 = \dfrac{1}{125}(125V) = V. \nonumber$$

Step by Step Process for Finding the Inverse

1. Interchange the variables
2. Solve for $y$
3. Write in terms of $f^{-1}(x)$

Example$\PageIndex{5}$

Find the inverse of

$$f (x) = y = 3x^3 - 5 \nonumber$$

Solution

$$\begin{align} x &= 3y^3 - 5 \\ x + 5 &= 3y^3 \\ \dfrac{(x + 5)}{3} &= y^3 ,\\ \left[\dfrac{(x + 5)}{3}\right]^{\frac{1}{3}}&=y \end{align}$$

$$f^{-1}(x) = \left[\dfrac{(x + 5)}{ 3 }\right]^{\frac{1}{3}}. \nonumber$$

Graphing

To graph an inverse we draw the $y = x$ line and reflect the graph across this line.

To interactively view the graph of an inverse click here:

mathcsjava.emporia.edu/~greenlar/Inverse/inverse.html

Contributors and Attributions

• Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.