OP - Inverses of Functions Lesson

Inverses of Functions

image of an inverse: f(x) / f to the -1(x) with arrows pointing to and from both equations In this topic, we will be looking at the inverse of a function. In order to do that, we must review what makes a function and the concept of one-to-one.

Remember the definition of a function - for every one number in the domain, there is one unique number in the range . In other words, each x can have only one y associated with it. You have used the vertical line test to determine if a graph is a function. The test states that if every vertical line intersects the graph at no more than one point, then it is a function.

What is a one-to-one function? This is a function in which each y also has only one x associated with it. In other words, one x relates only to one y and one y relates only to one x. The test to determine one-to-one is the horizontal line test. The horizontal line test requires that every horizontal line intersect the graph at no more than one point.

Only functions that are one-to-one have an inverse. For the inverse of a function, the domain and range values switch creating a new "function." If f(x) has the points (1, 4), (2, 5) and (3, 6), then the inverse function, denoted f(x)^-1 will have the points (4, 1), (5, 2), and (6, 3).

With a function in equation form, find the inverse by switching x and y and then solving for y.

Example of Finding the Inverse Function One

Function f(x) = 2x + 3 can be written y = 2x + 3

Switch the x and y: x = 2y + 3

Solve for y: equation image indicator $LaTeX: x-3=2y\: \\ y=\frac{x-3}{2}\\ y= \frac{1}{2}x -\frac{3}{2}$ $x-3=2y\: \\ y=\frac{x-3}{2}\\ y= \frac{1}{2}x -\frac{3}{2}$

Write in inverse notation: equation image indicator $LaTeX: f^{-1}\left(x\right)=\frac{1}{2}x-\frac{3}{2}$ $f^{-1}\left(x\right)=\frac{1}{2}x-\frac{3}{2}$

Example of Finding the Inverse Function Two

equation image indicator $LaTeX: f\left(x\right) =\frac{1}{x},\:x\ne0 \\ Let: y= \frac{1}{x} \\ Switch\; x \;and\; y: x= \frac{1}{y} \\ Solve for \;y: y= \frac{1}{x} \\ So, f^{-1} (x)=\frac{1}{x}, x\ne 0$ $f\left(x\right) =\frac{1}{x},\:x\ne0 \\ Let: y= \frac{1}{x} \\ Switch\; x \;and\; y: x= \frac{1}{y} \\ Solve for \;y: y= \frac{1}{x} \\ So, f^{-1} (x)=\frac{1}{x}, x\ne 0$

This is an example of a self-inverse function: equation image indicator $LaTeX: f^{-1}\left(x\right)=f\left(x\right)$ $f^{-1}\left(x\right)=f\left(x\right)$ .

Cryptography

Inverse functions are used by government agencies and other businesses to encode and decode information. These functions are usually very complicated. A simplified example involves the function equation image indicator . If each letter of the alphabet is assigned a numerical value according to its position (A = 1, B = 2, ..., Z = 26), the word ALGEBRA would be encoded by putting the numbers for each letter into the function, getting 1 34 19 13 4 52 1. The "message" can be decoded by finding the inverse function and plugging the encoded numbers in to find the numbers corresponding to the letters.

What is the inverse of this function?
- Solution: $LaTeX: f\left(x\right)=\frac{\left(x+2\right)}{3}$ $f\left(x\right)=\frac{\left(x+2\right)}{3}$
What numbers do you get when you put the encoded number into the inverse?
- Solution: 1, 12, 7, 5, 2, 18, 1
What are the letters that match these numbers?
- Solution: A L G E B R A

Now it is time for you to complete the Inverses Self-Assessment.

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