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Mathematics LibreTexts

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.

To interactively view the graph of an inverse click here:

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

Contributors

  • Integrated by Justin Marshall.