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.