Let's evaluate the functions step by step at the indicated values:

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Exercise 5: $f(x) = -2x^2 + 3x$

(a) $f(-3)$:
Substitute $x = -3$ into $f(x)$: $f(-3) = -2(-3)^2 + 3(-3)$ $f(-3) = -2(9) - 9 = -18 - 9 = -27$

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(b) $f(2)$:
Substitute $x = 2$ into $f(x)$: $f(2) = -2(2)^2 + 3(2)$ $f(2) = -2(4) + 6 = -8 + 6 = -2$

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(c) $f(-a)$:
Substitute $x = -a$ into $f(x)$: $f(-a) = -2(-a)^2 + 3(-a)$ $f(-a) = -2(a^2) - 3a = -2a^2 - 3a$

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(d) $-f(a)$:
First find $f(a)$ by substituting $x = a$ into $f(x)$: $f(a) = -2a^2 + 3a$ Now multiply by $-1$: $-f(a) = -(-2a^2 + 3a) = 2a^2 - 3a$

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(e) $f(a+h)$:
Substitute $x = a+h$ into $f(x)$: $f(a+h) = -2(a+h)^2 + 3(a+h)$ First expand $(a+h)^2$: $(a+h)^2 = a^2 + 2ah + h^2$ Now substitute: $f(a+h) = -2(a^2 + 2ah + h^2) + 3(a+h)$ Distribute: $f(a+h) = -2a^2 - 4ah - 2h^2 + 3a + 3h$ Thus: $f(a+h) = -2a^2 - 4ah - 2h^2 + 3a + 3h$

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Exercise 6: $f(x) = 2|3x - 1|$

(a) $f(-3)$:
Substitute $x = -3$ into $f(x)$: $f(-3) = 2|3(-3) - 1|$ $f(-3) = 2| -9 - 1| = 2| -10 | = 2(10) = 20$

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(b) $f(2)$:
Substitute $x = 2$ into $f(x)$: $f(2) = 2|3(2) - 1|$ $f(2) = 2|6 - 1| = 2|5| = 2(5) = 10$

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(c) $f(-a)$:
Substitute $x = -a$ into $f(x)$: $f(-a) = 2|3(-a) - 1|$ $f(-a) = 2| -3a - 1 |$

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(d) $-f(a)$:
First find $f(a)$ by substituting $x = a$ into $f(x)$: $f(a) = 2|3a - 1|$ Now multiply by $-1$: $-f(a) = -2|3a - 1|$

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(e) $f(a+h)$:
Substitute $x = a+h$ into $f(x)$: $f(a+h) = 2|3(a+h) - 1|$ Simplify inside the absolute value: $f(a+h) = 2|3a + 3h - 1|$

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Final Answers Summary:

Exercise 5: $f(x) = -2x^2 + 3x$

1. $f(-3) = -27$
2. $f(2) = -2$
3. $f(-a) = -2a^2 - 3a$
4. $-f(a) = 2a^2 - 3a$
5. $f(a+h) = -2a^2 - 4ah - 2h^2 + 3a + 3h$

Exercise 6: $f(x) = 2|3x - 1|$

1. $f(-3) = 20$
2. $f(2) = 10$
3. $f(-a) = 2| -3a - 1 |$
4. $-f(a) = -2|3a - 1|$
5. $f(a+h) = 2|3a + 3h - 1|$

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Related Questions:

1. How do you simplify expressions involving absolute values like $|3x - 1|$?
2. What are the properties of quadratic functions like $-2x^2 + 3x$?
3. How do you expand $(a+h)^2$ step by step?
4. How does multiplying by $-1$ affect a function like $-f(a)$?
5. How do you evaluate composite functions, such as $f(g(x))$?

Tip: Always distribute carefully when expanding squared binomials and keep track of signs!