Suppose that the functions fand g are defined as follows. f(x)=x^2+7 g(x)= 2/5x , x!= 0 Find the compositions fof and g°g. Simplify your answers as much as possible. (Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.) (fcirc f)(x)=□ □ /□ × (gcirc g)(x)=□

To solve the problem, let's first determine the compositions \( f \circ f \) and \( g \circ g \) by substituting the functions into each other.

### 1. Finding \( f \circ f \):

Given:
\[ f(x) = x^2 + 7 \]

The composition \( f \circ f \) means applying \( f(x) \) to itself:
\[ (f \circ f)(x) = f(f(x)) \]

Substitute \( f(x) \) into itself:
\[ f(f(x)) = f(x^2 + 7) \]

Now, replace \( x \) in \( f(x) = x^2 + 7 \) with \( x^2 + 7 \):
\[ f(x^2 + 7) = (x^2 + 7)^2 + 7 \]

Expand \( (x^2 + 7)^2 \):
\[ (x^2 + 7)^2 = x^4 + 14x^2 + 49 \]

So:
\[ f(x^2 + 7) = x^4 + 14x^2 + 49 + 7 \]

Simplify the expression:
\[ f(f(x)) = x^4 + 14x^2 + 56 \]

Thus:
\[ (f \circ f)(x) = x^4 + 14x^2 + 56 \]

### 2. Finding \( g \circ g \):

Given:
\[ g(x) = \frac{2}{5}x \quad \text{where } x \neq 0 \]

The composition \( g \circ g \) means applying \( g(x) \) to itself:
\[ (g \circ g)(x) = g(g(x)) \]

Substitute \( g(x) \) into itself:
\[ g(g(x)) = g\left(\frac{2}{5}x\right) \]

Now, replace \( x \) in \( g(x) = \frac{2}{5}x \) with \( \frac{2}{5}x \):
\[ g\left(\frac{2}{5}x\right) = \frac{2}{5} \left(\frac{2}{5}x\right) \]

Simplify the expression:
\[ g(g(x)) = \frac{4}{25}x \]

Thus:
\[ (g \circ g)(x) = \frac{4}{25}x \]

### Final Answers:
- \( (f \circ f)(x) = x^4 + 14x^2 + 56 \)
- \( (g \circ g)(x) = \frac{4}{25}x \)

Would you like more details on any of these steps, or have any other questions?

Here are 5 related questions:
1. How do you find the domain of a composed function?
2. What happens to the domain of \( f(x) \) when it’s composed with itself?
3. Can \( g(x) \) be composed with itself if \( x = 0 \)?
4. How does composition differ from function multiplication?
5. What is the difference between \( f \circ g \) and \( g \circ f \)?

**Tip:** When composing functions, always ensure that the output of the first function falls within the domain of the second function.

Explore how to find the compositions f(f(x)) and g(g(x)) step-by-step. Understand the application of functions, including quadratic and rational functions, in composite forms. Suitable for high school students and anyone interested in advanced algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation