Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of g, of, f, of, minus, 3, .g(f(−3)).
f, of, x, equals, minus, x, minus, 7
f(x)=
−x−7
g, of, x, equals, 2, x, squared, minus, 3, x, plus, 15
g(x)=
2x
2
−3x+15

To find the value of \( g(f(-3)) \), we first need to evaluate \( f(-3) \) using the function \( f(x) = -x - 7 \), and then substitute that result into the function \( g(x) = 2x^2 - 3x + 15 \).

### Step 1: Evaluate \( f(-3) \)
Using the function \( f(x) = -x - 7 \):

\[
f(-3) = -(-3) - 7 = 3 - 7 = -4
\]

So, \( f(-3) = -4 \).

### Step 2: Substitute \( f(-3) \) into \( g(x) \)
Now, substitute \( f(-3) = -4 \) into the function \( g(x) = 2x^2 - 3x + 15 \):

\[
g(-4) = 2(-4)^2 - 3(-4) + 15
\]

### Step 3: Simplify \( g(-4) \)
First, calculate each term:

\[
2(-4)^2 = 2 \times 16 = 32
\]
\[
-3(-4) = 12
\]

Now sum all terms:

\[
g(-4) = 32 + 12 + 15 = 59
\]

### Final Answer:
\[
g(f(-3)) = 59
\]

Would you like further details or have any questions?

Here are five related questions to expand on this:
1. What is the general approach for solving composite functions like \( g(f(x)) \)?
2. How does the domain of \( f(x) \) and \( g(x) \) affect the composite function?
3. How would the solution change if \( f(x) \) was a different linear function?
4. What happens if you switch the order and compute \( f(g(x)) \) instead?
5. Can we always compute \( g(f(x)) \) if \( f(x) \) and \( g(x) \) are defined differently?

**Tip:** Always solve the innermost function first when dealing with composite functions.

Given the definitions of f(x) = -x - 7 and g(x) = 2x^2 - 3x + 15, find the value of g(f(-3)).

Learn how to solve a composite function g(f(x)) where f(x) = -x - 7 and g(x) = 2x^2 - 3x + 15. This step-by-step solution covers algebraic concepts like substitution and quadratic equations, making it suitable for students in Grades 9-10.

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