MM - Inverse Functions Lesson

Inverse Functions

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 equation image indicator $LaTeX: f^{-1}\left(x\right)$ $f^{-1}\left(x\right)$ will have the points (4, 1), (5, 2), and (6, 3).

Function - f(x)	Inverse Function - f^-1(x)
Domain - 1, 2, 3	Domain - 4, 5, 6
Range - 4, 5, 6	Range - 1, 2, 3

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

Watch the following videos that will review and explore inverses of functions.

Example

if f(x)=2x+3 can be written as y-2x+3
x=2y+3, switch the x and y
x=2y+3
x-3=2y
y=(1/2)x-(3/2)
solve for y
f to the -1 times (x)=(1/2)x-(3/2)

To verify that functions are inverses, show that equation image indicator $LaTeX: f^{-1}\left(f\left(x\right)\right)=x\:and\:f\left(f^{-1}\left(x\right)\right)=x$ $f^{-1}\left(f\left(x\right)\right)=x\:and\:f\left(f^{-1}\left(x\right)\right)=x$

Example

Using the function and inverse above-

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

Note - "Verify" is essentially a proof, so you must include each step as you simplify.

Watch the following videos that will review and explore verifying inverse of functions.

Graphs of Inverse Functions

The graphs of inverse functions are symmetric across the line y = x .

Using the function and inverse above, the graph is-

graph of inverse functions with three lines that intersection at (-3, -3)

Also recall that exponential functions and logarithmic functions are inverses.

inverses of logarithmic functions y=2 to the x, line, y=logsub2x

Here is an example of finding the inverse of a logarithmic function.

f(x)=logsub2(x+1)
can be written has
y=logsub2(x+1)

x=logsub2(y+1)
switch the x and y

2 to x=y+1
y=2 to x-1
solve for y

f to -1 times (x)=2 to x -1
write in inverse notation

Many functions don't have inverses, though sometimes the domain can be restricted to a section of the graph that is one-to-one.

Let's see how this is used.

Watch the following videos that will review and explore graphing inverses of functions.

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 $LaTeX: f\left(x\right)=3x-2$ $f\left(x\right)=3x-2$ . 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.

a. What is the inverse of this function?

Solution: $LaTeX: f^{-1}\left(x\right)=\left(x+2\right)/\:3$ $f^{-1}\left(x\right)=\left(x+2\right)/\:3$

b. What numbers do you get when you put the encoded number into the inverse?

Solution: 1, 12, 7, 5, 2, 18, 1

c. What are the letters that match these numbers?

Solution: A L G E B R A

It's now time for us to explore and practice working Composition of Functions.

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