https://file.aiquickdraw.com/mathbot/file/17265775724692zc0x729.jpg

Let's solve the problems given in the image.

The functions provided are:

- \( f(x) = 4x - 5 \)
- \( g(x) = x^2 + 4 \)

We are asked to compute the following:

### a) \( (f \circ g)(x) \) or \( f(g(x)) \)
This means we need to substitute \( g(x) \) into \( f(x) \).

\[
f(g(x)) = f(x^2 + 4) = 4(x^2 + 4) - 5
\]
\[
= 4x^2 + 16 - 5
\]
\[
= 4x^2 + 11
\]

### b) \( (g \circ f)(x) \) or \( g(f(x)) \)
This means we need to substitute \( f(x) \) into \( g(x) \).

\[
g(f(x)) = g(4x - 5) = (4x - 5)^2 + 4
\]
\[
= (16x^2 - 40x + 25) + 4
\]
\[
= 16x^2 - 40x + 29
\]

So the answers are:

- \( (f \circ g)(x) = 4x^2 + 11 \)
- \( (g \circ f)(x) = 16x^2 - 40x + 29 \)

Would you like any further details or explanations?

Here are some related questions for practice:
1. What is \( (f \circ f)(x) \) if \( f(x) = 4x - 5 \)?
2. What is \( (g \circ g)(x) \) if \( g(x) = x^2 + 4 \)?
3. If \( f(x) = 2x + 3 \) and \( g(x) = x^2 - 1 \), what is \( (f \circ g)(x) \)?
4. How would you find \( f(x + 1) \) for \( f(x) = 4x - 5 \)?
5. How does the result of \( (g \circ f)(x) \) change if \( g(x) = x^2 - 4 \)?

**Tip:** When dealing with composite functions, always apply the inner function first, then the outer function.

Solve the following composite function problems:
1. (f ∘ g)(x) with f(x) = 4x - 5 and g(x) = x^2 + 4
2. (g ∘ f)(x) with f(x) = 4x - 5 and g(x) = x^2 + 4

Learn how to solve composite functions with f(x) = 4x - 5 and g(x) = x^2 + 4. This example walks through the steps to find (f ∘ g)(x) and (g ∘ f)(x), showcasing key algebraic concepts and composite function applications for high school students.

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