Verifying inverse functions by composition | Mathematics III | High School Math | Khan Academy

TL;DR

By evaluating composite functions and finding inverses, it is discovered that f(g(x)) = x and g(f(x)) = x.

Transcript

• [Voiceover] Let's say that f of x is equal to x plus 7 to the third power, minus one. And let's say that g of x g of x is equal to the cube root of x plus one the cube root of x plus one, minus seven. Now, what I wanna do now is evaluate f of g of x I wanna evaluate f of g of x and I also wanna evaluate g of f of x g of f of x, and see what I get... Read More

Key Insights

• ❓ Evaluating composite functions involves substituting the output of one function into another.
• ☺️ f(g(x)) simplifies to x, showing that f(x) and g(x) are inverses of each other.
• ☺️ g(f(x)) also simplifies to x, further confirming that f(x) and g(x) are inverses.

Questions & Answers

Q: What does it mean to evaluate a composite function?

Evaluating a composite function involves substituting the output of one function as the input for another. In this case, we evaluate f(g(x)) and g(f(x)).

Q: What is the result of evaluating f(g(x))?

By substituting g(x) into the function f(x), f(g(x)) simplifies to x. This means that the composite function f(g(x)) results in the original input value x.

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

To evaluate g(f(x)), we substitute f(x) into the function g(x). This simplifies g(f(x)) to x, showing that the composite function g(f(x)) also results in the original input value.

Q: What is the relationship between f(g(x)) and g(f(x))?

The video demonstrates that both f(g(x)) and g(f(x)) simplify to x. This means that f(x) and g(x) are inverses of each other, as applying one function and then the other in either order results in the original input value x.

Summary & Key Takeaways

• The video demonstrates how to evaluate composite functions f(g(x)) and g(f(x)).

• By substituting g(x) into the function f(x), f(g(x)) simplifies to x.

• Similarly, by substituting f(x) into the function g(x), g(f(x)) also simplifies to x.