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# 1.4: Composition and Inverses

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## 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.