$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1.4: Composition and Inverses

### 1. Composition of Functions

Example 1:

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

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

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

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

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

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

Exercise

If
$$f(x) = 3x + 2$$

$$g(x) = 2x^2 + 1$$

$$h(x) = \sqrt{x-2}$$

$$c(x) = 4$$

Find $$f(g(x))$$

Find $$f(h(x))$$

Find $$f(f(x))$$

Find $$h(c(x))$$

Find $$c(f(g(h(x))))$$

### 2. 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 2:

If

$f(x) = 3x + 1$

then

$3a + 1 = 3b + 1$

implies that

$3a = 3b$

hence

$a = b$

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

Example 3:

If

$f(x) = x^2$

then

$a^2 = b^2$

implies that

$a^2-b^2 = 0$

or that

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

hence

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

For example

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

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

### 3. Horizontal Line Test

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

### 4. Inverse Functions

Definition: Inverse function

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

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

Example 4:

The volume of a lake is modeled by the equation

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

Show that the inverse is

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

Solution: We have

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

and

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

### 5. 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 5:

Find the inverse of

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

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}

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

### 6. Graphing:

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