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 $\PageIndex{1}$

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

Find

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

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 $\PageIndex{2}$

\[f(x) = 3x + 1 \nonumber\]

then

\[3a + 1 = 3b + 1 \nonumber\]

implies that

\[3a = 3b \nonumber\]

hence

\[a = b \nonumber\]

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

Example $\PageIndex{3}$

\[f(x) = x^2 \nonumber\]

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.

$oneone.gif$

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

Interchange the variables
Solve for $y$
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.

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