10: Inverse Functions
Objectives
After completing this lesson, you should be able to:
- Determine if a function is one-to-one
- Find the inverse of a function
- Graph functions and their inverses
Instructor Commentary
In Lesson 8, we learned about two functions that were inverses of each other. Not all functions have inverses. In order for a function to have an inverse, it must be a one-to-one function. This means it passes a vertical line test to be a function, but then it must also pass a horizontal line test to be considered one-to-one.
In general, for every x-value in the domain, there is one and only one unique associated y-value in the range. No two x-values can have the same associated y-value. This allows us to switch the domain and range to generate the points from the function to get the points for the inverse function. Below are examples of functions that are one-to-one and functions that are not one-to-one.
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Functions that are NOT one-to-one
- A horizontal line is shown crossing the graphs in more than one place
- Two values in the domain are associated with the same value in the range
- These are functions because they satisfy the vertical line test
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One-to-One Functions
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Given a one-to-one function, we can find the inverse algebraically following these two steps:
- Switch the variables x and y
- Solve for y to find the inverse function
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Example
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Find the inverse of the function g(x) = 3x + 6
This is the same as saying y = 3x + 6, so we begin by switching the x and y variables (wherever there is an x, make it a y and wherever there is a y, make it an x).
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x = 3y + 6
Now we solve for y
x – 6 = 3y
 x – 2 = y
So g-1(x) =  x – 2
Note the points on the two functions
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Graphing the Function and the Inverse on the same grid.
One is the reflection of the other over the line y = x
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 g(x) = 3x + 6
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x
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y
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- 3
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- 3
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- 2
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0
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- 1
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3
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0
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6
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1
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9
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g-1(x) = x – 2
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x
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y
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- 3
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- 3
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0
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- 2
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3
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- 1
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6
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0
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9
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1
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You can check your new function. When you place the inverse of a function inside of the function you get x and when you place the function inside of the inverse you also get x.
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Example
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Find the inverse of the function h(x) = 
y =
Switch x and y
x = 
Solve for y
x(y + 1) = 2y – 4
xy + x = 2y – 4
xy – 2y = - x – 4
y(x – 2) = - x – 4
y = 
h-1(x) =
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You can check your new function. When you place the inverse of a function inside of the function you get x and when you place the function inside of the inverse you also get x.
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Example
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Find the inverse of the function k(x) = x2 for x ≤ 0
Note: x 2 is NOT a one-to-one function, but if we restrict the domain to only consider part of the function, then we can MAKE the function one-to-one
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y = x2 for x ≤ 0
Switch x and y
x = y2 for y ≤ 0
± = y
Since we have y ≤ 0 there is no confusion as to whether we take the ± root. We need to take the negative root since y ≤ 0
y = -
k-1(x) = -
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You can check your new function. When you place the inverse of a function inside of the function you get x and when you place the function inside of the inverse you also get x.
The x values we are considering are negative values so to make sure the square root is positive we have the opposite of those x values
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Example
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Find the inverse of the function f(x) = x2 for x ≥ 0
y = x2 for x ≥ 0
Switch x and y
x = y2 for y ≥ 0
± = y
Since we have y ≥ 0 there is no confusion as to whether we take the ± root. We need to take the positive root since y ≥ 0
y =
f-1(x) =
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You can check your new function. When you place the inverse of a function inside of the function you get x and when you place the function inside of the inverse you also get x.
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