let f(x)=2x^2+6 and g(x)= -3x+3. Find the following: (f-g)(-2)

To solve \((f - g)(-2)\), we'll first break this down into steps.

### Step 1: Express \((f - g)(x)\)
The operation \((f - g)(x)\) means subtracting the function \(g(x)\) from \(f(x)\):
\[
(f - g)(x) = f(x) - g(x)
\]
Given:
\[
f(x) = 2x^2 + 6
\]
\[
g(x) = -3x + 3
\]
Thus:
\[
(f - g)(x) = (2x^2 + 6) - (-3x + 3)
\]
Simplifying:
\[
(f - g)(x) = 2x^2 + 6 + 3x - 3
\]
\[
(f - g)(x) = 2x^2 + 3x + 3
\]

### Step 2: Evaluate \((f - g)(-2)\)
Now substitute \(x = -2\) into \((f - g)(x)\):
\[
(f - g)(-2) = 2(-2)^2 + 3(-2) + 3
\]
First, calculate each term:
\[
2(-2)^2 = 2(4) = 8
\]
\[
3(-2) = -6
\]
So:
\[
(f - g)(-2) = 8 - 6 + 3
\]
\[
(f - g)(-2) = 5
\]

### Final Answer:
\[
(f - g)(-2) = 5
\]

Would you like further details or have any questions?

Here are some related questions you can explore:
1. What is \((f + g)(x)\) for the same functions?
2. How would you find \((f \cdot g)(x)\) at \(x = 1\)?
3. What is the value of \(g(f(x))\) when \(x = 0\)?
4. Can you find where \(f(x) = g(x)\)?
5. What is the derivative of \((f - g)(x)\)?

**Tip:** When combining functions, simplify the expressions carefully before substituting values!

Learn how to solve (f - g)(-2) for the functions f(x) = 2x^2 + 6 and g(x) = -3x + 3. This detailed step-by-step solution involves function operations, quadratic functions, and simplifying expressions. Suitable for students in Grades 9-11.

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