RANGE (-2, 4 ) HORIZONTAL LINE y = x One-to-one x 5 x. If f and g are inverse functions , then ( f o g ) (x) = 1._______ and ( g o f ) ( x).

It looks like you're asking about properties of inverse functions, specifically about the composition of functions \( f \) and \( g \) when they are inverses. Let's break this down step by step:

### Key Concept:
If \( f(x) \) and \( g(x) \) are inverse functions, then:
- \( (f \circ g)(x) = f(g(x)) = x \)
- \( (g \circ f)(x) = g(f(x)) = x \)

In other words, applying \( g \) and then \( f \) (or \( f \) and then \( g \)) returns the original input \( x \). 

### Filling in the Blanks:
1. **If \( f \) and \( g \) are inverse functions, then \( (f \circ g)(x) = x \).**
2. Similarly, \( (g \circ f)(x) = x \).**

This property is essential for verifying that two functions are indeed inverses.

Would you like more details, or do you have any questions?

Here are five related questions to explore further:
1. How do you check if two functions are inverses using composition?
2. What is the graphical relationship between a function and its inverse?
3. How can the horizontal line test help determine if a function has an inverse?
4. What are the implications of a function being one-to-one?
5. How do you find the inverse of a function algebraically?

**Tip:** If a function passes the horizontal line test, it is one-to-one, meaning it has an inverse function.

Explore the properties of inverse functions and composition in mathematics. Learn how to verify if functions f and g are inverses, and understand the significance of the compositions f o g(x) = x and g o f(x) = x. Suitable for high school students studying advanced algebra.

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