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Section0.3Transformations, Compositions, and Inverses

Objectives
  • How can new functions be generated by shifts, stretches, and transformations of well-known functions?

  • How can we mathematically describe symmetric functions?

  • How can we build inverse functions, and when do those functions exist?

There are infinitely many functions that can be generated using the basic mathematical operations (addition, subtraction, multiplication, division, and exponentiation) along with simple functions such as roots, exponentials, and trigonometric functions. In fact, we can build entire families of functions based only on these simple building blocks.

Preview Activity0.3.1

The goal of this activity is to explore and experiment with the function

\begin{equation*} F(x) = Af(B(x-C))+D. \end{equation*}

The values of \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) are constants and the function \(f(x)\) will be henceforth called the parent function. To facilitate this exploration, use the applet located at

http://www.geogebratube.org/student/m93018.

  1. Let's start with a simple parent function: \(f(x) = x^2\text{.}\)

  2. Fix \(B=1\text{,}\) \(C=0\text{,}\) and \(D=0\text{.}\) Write a sentence or two describing the action of \(A\) on the function \(F(x)\text{.}\)

  3. Fix \(A=1\text{,}\) \(B=1\text{,}\) and \(D=0\text{.}\) Write a sentence of two describing the action of \(C\) on the function \(F(x)\text{.}\)

  4. Fix \(A=1\text{,}\) \(B=1\text{,}\) and \(C=0\text{.}\) Write a sentence of two describing the action of \(D\) on the function \(F(x)\text{.}\)

  5. Fix \(A=1\text{,}\) \(C=0\text{,}\) and \(D=0\text{.}\) Write a sentence of two describing the action of \(B\) on the function \(F(x)\text{.}\)

  6. In part (a) you have made conjectures about what \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) do to a parent function graphically. Test your conjectures with the functions \(f(x) = |x|\) (typed abs(x)), \(f(x) = x^3\text{,}\) \(f(x) = \sin(x)\text{,}\) \(f(x) = e^x\) (typed exp(x)), and any other function you find interesting.

Subsection0.3.1Function Transformations

In Preview Activity 1 we experimented with the four main types of function transformations. You no doubt noticed that the values of \(C\) and \(D\) shift the parent function and the values of \(A\) and \(B\) stretch the parent function.

Function Transformations
If \(f(x)\) is a parent function and

\begin{equation*} F(x) = Af(B(x-C))+D \end{equation*}

then the actions of each parameter are described in the table below.

Parameter Action
\(A\) Stretch the parent function vertically
\(B\) Stretch the parent function horizontally
\(C\) Shift the parent function horizontally
\(D\) Shift the parent function vertically
Table0.3.1Actions of stretch- and shift-type transformations

Example 0.3.2

Consider the function \(f(x)\) in the left-hand plot of the figure below. Plot the functions \(g(x) = 2f(x)\text{,}\) \(j(x) = f(2x)\text{,}\) \(h(x) = f(x)+1\text{,}\) and \(k(x) = f(x-1)\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

Figure0.3.3A function with transformations.
Solution
Activity0.3.2

Consider the function \(f(x)\) displayed in the figure below.

  1. Plot \(g(x) = -f(x)\) and \(h(x) = f(x)-1\).

  2. Define the function \(k(x) = -f(x)-1\). Does it matter which order you complete the tranformations from part (a) to result in \(k(x)\)? Plot the functions resulting from doing the two transformation in part (a) in opposite orders. Which of these functions is \(k(x)\)?

Hint
Solution

Subsection0.3.2Composition of Functions

When multiple transformations are applied in sequence, like in the previous activity, the resulting function is actually the composition of function transformations. The concept of a composition encompasses more than just transformations though. If \(f(x)\) and \(g(x)\) are functions where the range of \(g(x)\) is a subset of the domain of \(f(x)\) we can form a new function \(h(x) = f(g(x))\text{.}\) This literally means that you are substituting \(g(x)\) in for every instance of the variable \(x\) in \(f(x)\text{.}\) For example, if \(f(x) = x^2\) and \(g(x) = \sin(x)\) then \(h(x) = f(g(x)) = \left( \sin(x) \right)^2\) and \(k(x) = g(f(x)) = \sin(x^2)\text{.}\)

Example 0.3.5

If \(f(x) = x^2\) and \(g(x) = x-1\) then find \(f(g(3))\text{,}\) \(g(f(3))\text{,}\) \(f(g(x))\text{,}\) and \(g(f(x))\text{.}\)

Solution
Activity0.3.3
  1. Let \(f(x) = x^2\) and \(g(x) = x+8\). Find the following: \begin{equation*} f(g(3)) = \underline{\hspace{1in}}, \quad g(f(3)) = \underline{\hspace{1in}}, \quad f(g(x)) = \underline{\hspace{1in}}, \end{equation*} \begin{equation*} g(f(x)) = \underline{\hspace{1in}}, \quad f(x)g(x) = \underline{\hspace{1in}} \end{equation*}

  2. Now let \(f(x)\) and \(g(x)\) be defined as in the table below. Use the data in the table to find the following compositions.

    \( x \) −3 −2 −1 0 1 2 3
    \(f(x)\) 3 1 −1 −3 −1 1 3
    \( g(x) \) −2 −1 0 1 0 1 2
    \begin{equation*} f(-3) = \underline{\hspace{1in}}, \quad g(3) = \underline{\hspace{1in}}, \end{equation*} \begin{equation*} f(g(-3)) = \underline{\hspace{1in}}, \quad f(g(f(-3))) = \underline{\hspace{1in}} \end{equation*}

  3. Now let \(f(x)\) and \(g(x)\) be defined as in the plots below. Use the plots to find the following compositions.

    \begin{equation*} f(1) = \underline{\hspace{1in}}, \quad g(2) = \underline{\hspace{1in}}, \quad g(f(1)) = \underline{\hspace{1in}} \end{equation*} \begin{equation*} f(g(1)) = \underline{\hspace{1in}}, \quad g(f(f(0))) = \underline{\hspace{1in}} \end{equation*}

Hint
Solution

Subsection0.3.3Symmetry

There are many ways that a function can be symmetric, but two important symmetries are (1) reflective symmetry over the \(y\)-axis, and (2) \(180^\circ\) rotational symmetry about the origin.

Even Functions

A function that has reflective symmetry over the \(y\)-axis is called an even function.

Odd Functions

A function with rotational symmetry about the origin is called an odd function.

The reasoning for these names will be evident after completing the next activity.

Activity0.3.4
  1. Based on symmetry alone, is \(f(x) = x^2\) an even or an odd function?

  2. Based on symmetry alone, is \(g(x) = x^3\) an even or an odd function?

  3. Find \(f(-x)\) and \(g(-x)\) and make conjectures to complete these sentences:

    • If a function \(f(x)\) is even then \(f(-x) =\).

    • If a function \(f(x)\) is odd then \(f(-x) =\).

    Explain why the composition \(f(-x)\) is a good test for symmetry of a function.
  4. Classify each of the following functions as even, odd, or neither. \begin{equation*} h(x) = \frac{1}{x}, \quad j(x) = e^x, \quad k(x) = x^2-x^4, \quad n(x) = x^3+x^2. \end{equation*}

  5. Each figure below shows only half of the function. Draw the left half so \(f(x)\) is even. Draw the left half so \(g(x)\) is odd. Draw the left half so \(h(x)\) is neither even nor odd.

Hint
Solution
Test for Even and Odd Functions

  • A function \(f(x)\) is called even if and only if \( f(-x) = f(x) \)
  • A function \(f(x)\) is called odd if and only if \( f(-x) = -f(x) \)

Subsection0.3.4Inverse Functions

We conclude this section by discussing an important question: If we know the action of a function is it possible to undo that action? This question can be rephrased by saying: If we know the output of a function can we tell exactly what the input was? The answer to these questions is that it depends on the type of function.

Consider, for example, the function \(f(x) = x^2\text{.}\) If we know that \(f(a) = 4\) do we the value of \(a\text{?}\) Of course not! It is obvious that \(f(2) = f(-2) = 4\text{,}\) so just by knowing the output of the function \(f(x) = x^2\) we cannot invert the function and find the input. What about the function \(g(x) = x^3\text{?}\) If we know that \(g(b) = 8\) then there is only one unique value of \(b\text{,}\) \(b=2\text{,}\) such that \(g(b) = 8\text{.}\) Therefore it seems like we can invert the cubic function.

The act of reversing the action of a function can be explored geometrically. Indeed, in Figure 8 we see that if we can simply switch the values of \(x\) and \(y\) we will get a plot that shows how to undo the action of a function. Geometrically, switching the role of the \(x\) and the \(y\) in the function is the same as reflecting over the line \(y=x\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

Figure0.3.8Inverse functions. If \((x,y)\) is on one end of one of the dashed segments, then \((y,x)\) is on the other side.

The question that remains is when an inverse function actually exists. This is the same as asking: “if I reflect over \(y=x\) is the end result a function?” The answer to this question is certainly “no” if the function is \(f(x) = x^2\) (as seen in the left-hand plot of Figure 9), but if we restrict the domain on \(f(x) = x^2\) to \(0 \le x \lt \infty\) then the result is a function (as seen in the right-hand plot of Figure 9). This leads us to the following results.

Horizontal Line Test

If a horizontal line passes through a function only once, then it has a unique inverse found by interchanging the \(x\) and the \(y\text{.}\)

Inverses via Reflection

The inverse of a function can be found geometrically by reflecting the graph of the function over the line \(y=x\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

Figure0.3.9The left-hand plot shows that after reflecting \(f(x)=x^2\) across \(y=x\) the result is not a function. The right-hand plot shows that under a restriction of the domain the result can be a function.
Example 0.3.10

Find the inverse of the following functions. If necessary, restrict the domain on the function so that the inverse exists.

  1. \(f(x) = x^2+1\)

  2. \(g(x) = ax+b\)

  3. \(h(x) = (2x+8)^3\)

Solution

Finally, to tie the ideas of composition and inverses together we observe that if the inverse of a function switches the roles of \(x\) and \(y\) then the composition \(f^{-1}(f(x))\) should simply give \(x\) back. The logical argument is as follows:

\begin{equation*} f \text{ maps } x \text{ to } y \text{ then } f^{-1} \text{ maps } y \text{ to } x \end{equation*}

That is,

\begin{equation*} f^{-1}(f(x)) = x. \end{equation*}

Similarly,

\begin{equation*} f^{-1} \text{ maps } x \text{ to } y \text{ then } f \text{ maps } y \text{ to } x \end{equation*}

which is written more compactly as

\begin{equation*} f(f^{-1}(x)) = x. \end{equation*}

These two equations provide a nice algebraic check when finding inverses.

Checking Inverse Functions

If \(f(x)\) has an inverse \(f^{-1}(x)\) then \begin{equation*} f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \end{equation*}

Activity0.3.5
  1. Find the inverse of each of the following functions by interchanging the \(x\) and \(y\) and solving for \(y\). Be sure to state the domain for each of your answers. \begin{equation*} y = \sqrt{x-1}, \quad y = -\frac{1}{3} x + 1, \quad y = \frac{x+4}{2x-5} \end{equation*}

  2. Verify that the functions \(f(x) = 3x-2\) and \(g(x) = \frac{x}{3} + \frac{2}{3}\) are inverses of each other by computing \(f(g(x))\) and \(g(f(x))\).

Hint
Solution

Subsection0.3.5Summary

  • A function can be transformed by \(F(x) = Af(B(x-C))+D\) where \(C\) and \(D\) shift the function and \(A\) and \(B\) stretch the function.

  • If \(f(-x) = f(x)\) then \(f\) is an even function.

  • If \(f(-x) = -f(x)\) then \(f\) is an odd function.

  • To find the inverse of a function we switch the roles of the \(x\) and \(y\) variables. Geometrically this is the same as reflecting over the line \(y=x\text{.}\) Occasionally it is essential to restrict the domain of the original function in order for the inverse to exist.

  • The composition of a function and its inverse is the original input:

    \begin{equation*} f(f^{-1}(x)) = x \text{ and } f^{-1}(f(x)) = x. \end{equation*}

Subsubsection0.3.6Exercises

1

The functions \(f(x)\) and \(g(x)\) are defined in the table below. Use these function values to answer the following questions.

\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(f(x)\) \(3\) \(1\) \(-1\) \(-3\) \(-1\) \(1\) \(3\)
\(g(x)\) \(-2\) \(-1\) \(0\) \(1\) \(0\) \(1\) \(2\)

(a) \(f(-3)\text{,}\) (b) \(g(3)\text{,}\) (c) \(f(g(-3))\text{,}\) (d) \(g(f(3))\text{,}\) (e) \(f(g(f(-3)))\)

(f) Write a list of value of \(f(-x)\) for \(x = -3, -2, \dots, 2, 3\text{.}\) Based on this list is \(f(x)\) an even function, an odd function, or neither?

(g) Repeat part (f) for \(g(x)\text{.}\)

2

Find the inverse of each of the following functions. If necessary state a restriction on the domain of \(f(x)\) so that the inverse actually exists.

  1. \(f(x) = (2x-3)^2\)

  2. \(g(x) = x^2 - 2x + 1\)

3

The plot on the left shows the function \(f(x)\) and the plot on the right shows \(g(x) = Af(B(x-C))+D\text{.}\) Find the appropriate values of \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

4

Use the function below to plot

(a) \(f(x)-3\text{,}\) (b) \(f(x+1)\text{,}\) (c) \(\frac{1}{2}f(x)\text{,}\) (d) \(-f(x)\text{,}\) and (e) \(\frac{1}{f(x)}\text{.}\)

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