Let's evaluate each function step by step based on the given information:

Given:

• $F(x) = 2x - 5$
• $g(x) = 7 - 3x$
• $h(x) = 4 - 3x + x^2$

1. $f(-5)$

Substitute $x = -5$ into $F(x) = 2x - 5$: $f(-5) = 2(-5) - 5 = -10 - 5 = -15$

2. $g(2)$

Substitute $x = 2$ into $g(x) = 7 - 3x$: $g(2) = 7 - 3(2) = 7 - 6 = 1$

3. $h(-3)$

Substitute $x = -3$ into $h(x) = 4 - 3x + x^2$: $h(-3) = 4 - 3(-3) + (-3)^2 = 4 + 9 + 9 = 22$

4. $g(h(-3))$

First, find $h(-3)$ (already calculated above): $h(-3) = 22$ Now substitute $x = 22$ into $g(x) = 7 - 3x$: $g(22) = 7 - 3(22) = 7 - 66 = -59$

5. $f(f(-2))$

First, find $f(-2)$: $f(-2) = 2(-2) - 5 = -4 - 5 = -9$ Now substitute $x = -9$ into $F(x) = 2x - 5$: $f(f(-2)) = f(-9) = 2(-9) - 5 = -18 - 5 = -23$

6. $f(a+b)$

Substitute $x = a + b$ into $F(x) = 2x - 5$: $f(a+b) = 2(a + b) - 5 = 2a + 2b - 5$

7. $g(g(x))$

First, find $g(x) = 7 - 3x$. Now substitute $g(x)$ into $g$ again: $g(g(x)) = g(7 - 3x) = 7 - 3(7 - 3x) = 7 - 21 + 9x = -14 + 9x$

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Final answers:

1. $f(-5) = -15$
2. $g(2) = 1$
3. $h(-3) = 22$
4. $g(h(-3)) = -59$
5. $f(f(-2)) = -23$
6. $f(a+b) = 2a + 2b - 5$
7. $g(g(x)) = -14 + 9x$

Would you like further clarification on any of these?

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Related Questions:

1. What is $h(g(x))$?
2. What is $f(f(f(-1)))$?
3. Can we evaluate $h(f(x))$ for a specific value of $x$?
4. How do you find the inverse of $g(x)$?
5. What is the general form of $f(x) + g(x)$?

Tip:

When substituting values into functions, simplify the inner functions first before proceeding to the outer ones (like with $g(h(x))$).