https://file.aiquickdraw.com/mathbot/file/17266407983858dna91ms.jpg

Let's break down the problem from the image:

Given functions:
- \( f(x) = x + 2 \)
- \( g(x) = \sqrt{x - 3} \)

We need to find the compositions:
1. \( (f \circ g)(x) = f(g(x)) \)
2. \( (g \circ f)(x) = g(f(x)) \)

### 1. Finding \( (f \circ g)(x) \):
\( f(g(x)) = f(\sqrt{x - 3}) \)

Substitute \( \sqrt{x - 3} \) into \( f(x) \):
\[ f(\sqrt{x - 3}) = \sqrt{x - 3} + 2 \]

So, 
\[ (f \circ g)(x) = \sqrt{x - 3} + 2 \]

### 2. Finding \( (g \circ f)(x) \):
\( g(f(x)) = g(x + 2) \)

Substitute \( x + 2 \) into \( g(x) \):
\[ g(x + 2) = \sqrt{(x + 2) - 3} \]
\[ = \sqrt{x - 1} \]

So, 
\[ (g \circ f)(x) = \sqrt{x - 1} \]

### Final Composed Functions:
1. \( (f \circ g)(x) = \sqrt{x - 3} + 2 \)
2. \( (g \circ f)(x) = \sqrt{x - 1} \)

Do you want more details or have any questions about these steps?

Here are 5 related questions for further practice:
1. What is the domain of \( (f \circ g)(x) \) and \( (g \circ f)(x) \)?
2. How do you find the composition \( (h \circ f)(x) \) for another function \( h(x) \)?
3. If \( f(x) = 2x \) and \( g(x) = x^2 + 1 \), what is \( (f \circ g)(x) \)?
4. How do you check if two functions are inverses using composition?
5. Can you compose three functions together, like \( (f \circ g \circ h)(x) \)?

**Tip:** When composing functions, always substitute the inner function into the outer function carefully, following the correct order of operations.

Consider the functions f(x) = x + 2 and g(x) = sqrt(x - 3). Determine each of the following: (f ∘ g)(x) and (g ∘ f)(x).

Learn how to find the composition of two functions: f(x) = x + 2 and g(x) = sqrt(x - 3). This algebra problem covers key concepts like function composition and working with square roots. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation