https://file.aiquickdraw.com/mathbot/file/1727452583317dguu4v9o.jpg

The problem in the image shows three functions:

1. \( f(x) = x + 2 \)
2. \( g(x) = 3 - 2x \)
3. \( h(x) = x^2 + 3x - 4 \)

The question asks to find \( h \circ f \circ g(x) \), which is the composition of these functions: \( h(f(g(x))) \).

Let’s break this down step by step:

### Step 1: Find \( g(x) \)
\[
g(x) = 3 - 2x
\]

### Step 2: Find \( f(g(x)) \)
Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(3 - 2x) = (3 - 2x) + 2 = 5 - 2x
\]

### Step 3: Find \( h(f(g(x))) \)
Now substitute \( f(g(x)) = 5 - 2x \) into \( h(x) \):
\[
h(f(g(x))) = h(5 - 2x) = (5 - 2x)^2 + 3(5 - 2x) - 4
\]

### Step 4: Simplify
Expand the terms:
\[
(5 - 2x)^2 = 25 - 20x + 4x^2
\]
\[
3(5 - 2x) = 15 - 6x
\]
Now combine:
\[
h(f(g(x))) = 25 - 20x + 4x^2 + 15 - 6x - 4
\]
Simplify the constants and like terms:
\[
h(f(g(x))) = 4x^2 - 26x + 36
\]

### Final Answer:
\[
h(f(g(x))) = 4x^2 - 26x + 36
\]

Let me know if you have any questions or would like further details!

### Related Questions:
1. What is the composition \( f(g(f(x))) \)?
2. How do you find the inverse of \( f(x) = x + 2 \)?
3. Can you explain how to graph \( h(x) = x^2 + 3x - 4 \)?
4. What is the domain and range of \( f(x) \), \( g(x) \), and \( h(x) \)?
5. How does the composition of functions affect their behavior?

### Tip:
Always substitute functions carefully and simplify one step at a time to avoid errors in composition.

Find the composition of functions h(f(g(x))). Given:
1. f(x) = x + 2
2. g(x) = 3 - 2x
3. h(x) = x^2 + 3x - 4

Learn how to solve the composition of functions h(f(g(x))) where f(x) = x + 2, g(x) = 3 - 2x, and h(x) = x^2 + 3x - 4. Follow a detailed step-by-step guide, suitable for students in Grades 9-11.

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