Verifying inverse functions by composition: not inverse | High School Math | Khan Academy

TL;DR

Learn how to evaluate composite functions and determine if two functions are inverses of each other.

Transcript

• [Voiceover] Let's say that f of x is equal to two x minus three, and g of x, g of x is equal to 1/2 x plus three. What I wanna do in this video is evaluate what f of g of x is, and then I wanna evaluate what g of f of x is. So first, I wanna evaluate f of g of x, and then I'm gonna evaluate the other way around. I'm gonna evaluate g of f of x. Bu... Read More

Key Insights

• ❓ Evaluating composite functions involves substituting the output of one function into another function.
• 🙈 The order of evaluating composite functions matters, as seen in the difference between f(g(x)) and g(f(x)).
• ☺️ If f(g(x)) does not equal x or g(f(x)) does not equal x, then the two functions are not inverses.

Questions & Answers

Q: How do you evaluate f(g(x))?

To evaluate f(g(x)), substitute g(x) into the function f(x) and simplify. For example, if f(x) = 2x - 3 and g(x) = (1/2)x + 3, then f(g(x)) would be 2((1/2)x + 3) - 3, which simplifies to x + 3.

Q: What is the process to evaluate g(f(x))?

To evaluate g(f(x)), substitute f(x) into the function g(x) and simplify. If f(x) = 2x - 3 and g(x) = (1/2)x + 3, g(f(x)) would be (1/2)(2x - 3) + 3, which simplifies to x + 3/2.

Q: How can you determine if two functions are inverses of each other?

If applying one function to the output of the other function does not result in the original input, then the functions are not inverses. In the given example, f(g(x)) did not equal x and g(f(x)) did not equal x, so they are not inverses.

Q: Can two functions be inverses of each other?

Yes, two functions can be inverses of each other if applying one function to the output of the other function gives back the original input. However, in the provided example, f(x) and g(x) are not inverses.

Summary & Key Takeaways

• The video explains how to evaluate f(g(x)) and g(f(x)) for two given functions f(x) and g(x).

• To find f(g(x)), substitute g(x) into the function f(x) and simplify the expression.

• To find g(f(x)), substitute f(x) into the function g(x) and simplify the expression.

• The video also emphasizes that if f(g(x)) and g(f(x)) do not produce the original input x, then the two functions are not inverses of each other.