1.4: Composition and Inverses

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Oct 6, 2021
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- 1.3: Shifting and Reflecting
- 1.5: The Plane

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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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Composition of Functions

1-1 Functions

Horizontal Line Test

Inverse Functions

Graphing

Contributors and Attributions

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