*f*(*x*)=2*x*2+3*x*−5
g, of, x, equals, 3, x, squared, minus, 10*g*(*x*)=3*x*2−10
h, of, x, equals, 2, x, plus, 4*h*(*x*)=2*x*+4

Find:
g, of, f, of, h, of, minus, 3*g*(*f*(*h*(−3)))

We are tasked with finding \( g(f(h(-3))) \), where the functions \( f(x) \), \( g(x) \), and \( h(x) \) are defined as follows:

- \( f(x) = 2x^2 + 3x - 5 \)
- \( g(x) = 3x^2 - 10 \)
- \( h(x) = 2x + 4 \)

Let's solve step by step.

### Step 1: Find \( h(-3) \)
We start with \( h(x) = 2x + 4 \).

Substitute \( x = -3 \):

\[
h(-3) = 2(-3) + 4 = -6 + 4 = -2
\]

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

\[
f(-2) = 2(-2)^2 + 3(-2) - 5 = 2(4) - 6 - 5 = 8 - 6 - 5 = -3
\]

### Step 3: Find \( g(f(h(-3))) \)
Substitute \( f(-2) = -3 \) into \( g(x) = 3x^2 - 10 \):

\[
g(-3) = 3(-3)^2 - 10 = 3(9) - 10 = 27 - 10 = 17
\]

### Final Answer:
\[
g(f(h(-3))) = 17
\]

Would you like a detailed breakdown of any part of the process, or have any questions?

Here are 5 related questions to extend your understanding:

1. What is the composition \( f(g(x)) \) with the same functions?
2. How do you compute \( h(f(g(x))) \)?
3. What happens when you evaluate \( g(h(-3)) \)?
4. Can you find \( g(f(h(0))) \)?
5. How would you interpret these compositions graphically?

**Tip:** When solving composite functions, always work from the innermost function outwards.

Find g(f(h(-3))) where f(x) = 2x^2 + 3x - 5, g(x) = 3x^2 - 10, and h(x) = 2x + 4.

Learn how to solve a composite function g(f(h(-3))) with quadratic and linear functions step-by-step. The problem involves finding h(-3), then f(h(-3)), and finally g(f(h(-3))). Suitable for high school students, Grades 10-12.

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