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Wize High School Grade 12 Pre-Calculus Textbook > Transformations

Inverse Functions & Transformations

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Inverse Functions & Transformations

An inverse function can be defined as a function that reverses another function.
  • f−1(x)f^{-1}(x) is the notation used to represent an inverse function
  • f−1(x) f^{-1}(x)~ is pronounded "f inverse x"
  • f(x) f(x)~ is reflected over the line y = x to get f−1(x)f^{-1}(x)
  • Swap 'x' and 'y' then solve for 'y' to find f−1(x)f^{-1}(x)

If two functions, f(x) and g(x), are inverses of each other, then the following two conditions must be true:
  • (f∘g)(x)=x(f\circ{g})(x)=x
  • (g∘f)(x)=x(g\circ{f})(x)=x
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Example 1
Let's find and graph f−1(x)f^{-1}(x) if f(x)=x2f(x)=x^2

Find f−1(x)f^{-1}(x):
Swap x & yx=y2Solve for yy2=xy=±x∴f−1(x)=±x\begin{array}{l c c l} \text{Swap x \& y}&&&x=y^2\\\\ \text{Solve for y}&&&y^2=x\\\\ &&&y=\pm\sqrt{x}\\\\ &&&\therefore f^{-1}(x)=\pm\sqrt{x} \end{array}
Table of values for f(x)=x2f(x)=x^2:
xy−24−11001124\begin{array}{|c|c|} \hline x&y\\\hline -2&4\\\hline -1&1\\\hline 0&0\\\hline 1&1\\\hline 2&4\\\hline \end{array}
Table of values for f−1(x)f^{-1}(x):
xy4−21−1001142\begin{array}{|c|c|} \hline x&y\\\hline 4&-2\\\hline 1&-1\\\hline 0&0\\\hline 1&1\\\hline 4&2\\\hline \end{array}

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Graph of f(x)  vs.  f−1(x)f(x)~~vs.~~f^{-1}(x):


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Example 2

Let's show that f(x)=2x−1f(x)=2x-1 and g(x)=12x+12g(x)=\frac{1}{2}x+\frac{1}{2} are inversers of each other.

f(g(x))f(g(x)):
f(g(x))=f(12x+12)=2(12x+12)−1=x+1−1=x\begin{array}{l c l} f(g(x))&=&f\Big(\frac{1}{2}x+\frac{1}{2}\Big)\\\\ &=&2\Big(\frac{1}{2}x+\frac{1}{2}\Big)-1\\\\ &=&x+1-1\\\\ &=&x \end{array}
g(f(x))g(f(x)):
g(f(x))=g(2x−1)=12(2x−1)+12=x−12+12=x\begin{array}{l c l} g(f(x))&=&g(2x-1)\\\\ &=&\frac{1}{2}(2x-1)+\frac{1}{2}\\\\ &=&x-\frac{1}{2}+\frac{1}{2}\\\\ &=&x \end{array}
Therefore, f(x) and g(x) are inverses of eachother

Example: Inverse Functions & Transformations

The table of values for y=f(x) y=f(x)~ is shown below:
xy−44−22216−3\begin{array}{|c|c|} \hline x&y\\\hline -4&4\\\hline -2&2\\\hline 2&1\\\hline 6&-3\\\hline \end{array}
What is the table of values for the following functions?
  1. y=2f−1(2x)+1y=2f^{-1}(2x)+1
  2. y=−f−1(−x−1)y=-f^{-1}(-x-1)

Part a.

y=2f−1(2x)+1y=2f^{-1}(2x)+1

First, find the table of values for f−1(x):f^{-1}(x):
xy4−42−212−36\begin{array}{|c|c|} \hline x&y\\\hline 4&-4\\\hline 2&-2\\\hline 1&2\\\hline -3&6\\\hline \end{array}
Then, apply the transformations:
  • Vertical expansion by 2 & 1 unit up
  • Horizontal compression by 12\frac{1}{2}

The table of values for y=2f−1(2x)+1y=2f^{-1}(2x)+1:
xy2−71−30.55−1.513\begin{array}{|c|c|} \hline x&y\\\hline 2&-7\\\hline 1&-3\\\hline 0.5&5\\\hline -1.5&13\\\hline \end{array}

Part b.

y=−f−1(−x−1)y=-f^{-1}(-x-1)

First, find the table of values for f−1(x)f^{-1}(x):
xy4−42−212−36\begin{array}{|c|c|} \hline x&y\\\hline 4&-4\\\hline 2&-2\\\hline 1&2\\\hline -3&6\\\hline \end{array}
Then, apply the transformations:
  • Vertically reflected about the x-axis
  • Horizontally reflected about the y-axis & 1 unit left

The table of values for y=−f−1(−x−1)y=-f^{-1}(-x-1):
xy−54−32−2−22−6\begin{array}{|c|c|} \hline x&y\\\hline -5&4\\\hline -3&2\\\hline -2&-2\\\hline 2&-6\\\hline \end{array}

Practice: Inverse Functions & Transformations

Let y=f(x) y=f(x)~ be graphed below:

Find and graph f−1(x)f^{-1}(x)

Practice: Inverse Functions & Transformations

The function y=f(x) y=f(x)~ is shown below:

Which graph represent y=−12f−1(2x−2)+4y=-\frac{1}{2}f^{-1}(2x-2)+4?

Practice: Inverse Functions & Transformations


If (1, 2) is on the graph of y=3f(2x), y=3f(2x),~ then what point must be on the function y=3f−1(2x)?y=3f^{-1}(2x)?
Extra Practice
Transformations
The graph of y=2(x−2)2−2y=2(x-2)^2-2 is shown below.
What is the inverse of y=2(x−2)2−2y=2(x-2)^2-2and its graph?
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
If (4a, 3b) is on the graph of y=−5f(x+2), y=-5f(x+2),~ then what point must be on the function y=−f−1(−x+1)?y=-f^{-1}(-x+1)?
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
If (3, -1) is on the graph of y=−2f(−x), y=-2f(-x),~ then what point must be on the function y=4f−1(x−2)?y=4f^{-1}(x-2)?
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
The function y=f(x) y=f(x)~ is shown below:
Which graph represent y=3f−1(x−4)+2y=3f^{-1}(x-4)+2?
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
The function y=f(x) y=f(x)~ has it's graph shown below:
Which graph represent y=2f−1(x+3)y=2f^{-1}(x+3)?
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)