To solve the problem:

We are given:

• $f(x) = 2x^2 + 5x - 7$
• $g(x) = 3x - 12$

We need to find $(g \circ f)(-1)$, which means $g(f(-1))$.

Step 1: Compute $f(-1)$

Substitute $x = -1$ into $f(x)$: $f(-1) = 2(-1)^2 + 5(-1) - 7$ $f(-1) = 2(1) - 5 - 7 = 2 - 5 - 7 = -10$

Step 2: Compute $g(f(-1)) = g(-10)$

Substitute $x = -10$ into $g(x)$: $g(-10) = 3(-10) - 12$ $g(-10) = -30 - 12 = -42$

Final Answer:

$(g \circ f)(-1) = -42$

Let me know if you'd like further clarification or have additional questions.

Here are 5 related questions to deepen your understanding:

1. How do we compute the composite function $(f \circ g)(x)$, and what is its result in this case?
2. How would the answer change if $f(x)$ and $g(x)$ were switched in composition?
3. What is the domain of the composite function $g \circ f(x)$?
4. How can you graphically represent $g \circ f(x)$?
5. What happens if we set $f(x) = g(x)$ and repeat the problem?

Tip: Always simplify functions step-by-step when working with composite functions