Lesson Objectives
- Learn how to determine if two functions are inverses
How to Determine if Two Functions are Inverses
How can we determine if two functions are inverses of each other? We previously stated that the domain of a function becomes the range of its inverse and the range of a function becomes the domain of its inverse. Let's take a simple function such as: $$f(x)=2x$$ This function maps an x-value of 5 to a y-value of 10. $$f(5)=2(5)=10$$ If we find the inverse of the function, it will map an x-value of 10 to a y-value of 5. $$f^{-1}(x)=\frac{x}{2}$$ $$f^{-1}(10)=\frac{10}{2}=5$$ We can think about the inverse as reversing the procedure of the original function. In the original function, we multiply 2 by an unknown number (x). In the inverse, we take the unknown number (x) and divide by 2. We know that multiplying by 2 and dividing by 2 are opposite operations and will cancel each other out. What happens if we plug in f-1(x) in for x in the original function? $$f(f^{-1}(x))=2\left(\frac{x}{2}\right)$$ $$\require{cancel}f(f^{-1}(x))=\cancel{2}\left(\frac{x}{\cancel{2}}\right)=x$$ Based on our example, we can state the following rule:
For inverses f and f-1: $$f(f^{-1}(x))=x$$ $$f^{-1}(f(x))=x$$ In other words, we can prove that two functions are inverses by plugging the inverse function in for x in the original function. If they are inverses, the results should just be x. We also need to check the other scenario. We will plug in the original function in for x in the inverse function. Again, if they are inverses, the result should just be x.
In other words, if f(x) and g(x) are inverses:
f(g(x)) = x
g(f(x)) = x
Let's look at an example.
Example 1: Determine if the functions f and g are inverses. $$f(x)=\sqrt[5]{x + 1}$$ $$g(x)=x^5 - 1$$ To determine if these two functions are inverses, let's plug g(x) in for x in f(x). $$f(g(x))=\sqrt[5]{(x^5 - 1) + 1}$$ $$f(g(x))=\sqrt[5]{x^5}$$ $$f(g(x))=x$$ Now we will plug f(x) in for x in g(x). $$g(f(x))=(\sqrt[5]{x + 1})^5 - 1$$ $$g(f(x))=x + 1 - 1$$ $$g(f(x))=x$$ In each case, we get x as a result. We can state that these two functions are inverses.
For inverses f and f-1: $$f(f^{-1}(x))=x$$ $$f^{-1}(f(x))=x$$ In other words, we can prove that two functions are inverses by plugging the inverse function in for x in the original function. If they are inverses, the results should just be x. We also need to check the other scenario. We will plug in the original function in for x in the inverse function. Again, if they are inverses, the result should just be x.
In other words, if f(x) and g(x) are inverses:
f(g(x)) = x
g(f(x)) = x
Let's look at an example.
Example 1: Determine if the functions f and g are inverses. $$f(x)=\sqrt[5]{x + 1}$$ $$g(x)=x^5 - 1$$ To determine if these two functions are inverses, let's plug g(x) in for x in f(x). $$f(g(x))=\sqrt[5]{(x^5 - 1) + 1}$$ $$f(g(x))=\sqrt[5]{x^5}$$ $$f(g(x))=x$$ Now we will plug f(x) in for x in g(x). $$g(f(x))=(\sqrt[5]{x + 1})^5 - 1$$ $$g(f(x))=x + 1 - 1$$ $$g(f(x))=x$$ In each case, we get x as a result. We can state that these two functions are inverses.
Skills Check:
Example #1
Determine if the two functions are inverses. $$f(x)=\sqrt[5]{x}- 2$$ $$g(x)=(x + 2)^5$$
Please choose the best answer.
A
Inverses
B
Not Inverses
Example #2
Determine if the two functions are inverses. $$f(x)=\frac{2}{3}x + \frac{2}{3}$$ $$g(x)=5x - 5$$
Please choose the best answer.
A
Inverses
B
Not Inverses
Example #3
Determine if the two functions are inverses. $$g(x)=\frac{2}{x - 1}+ 1$$ $$f(x)=\frac{2}{x - 1}+ 1$$
Please choose the best answer.
A
Inverses
B
Not Inverses
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