Find the following for the function f left parenthesis x right parenthesis equals 3 x squared plus 2 x minus 2.

​(a) f left parenthesis 0 right parenthesis
​(b) f left parenthesis 4 right parenthesis
​(c) f left parenthesis negative 4 right parenthesis
​(d) f left parenthesis negative x right parenthesis
​(e) negative f left parenthesis x right parenthesis
​(f)​ f(xplus2​)
​(g) ​f(3​x)
​(h) f left parenthesis x plus h right parenthesis

Let's evaluate the function \( f(x) = 3x^2 + 2x - 2 \) for each part:

### (a) \( f(0) \)

Substitute \( x = 0 \):

\[
f(0) = 3(0)^2 + 2(0) - 2 = -2
\]

### (b) \( f(4) \)

Substitute \( x = 4 \):

\[
f(4) = 3(4)^2 + 2(4) - 2 = 3(16) + 8 - 2 = 48 + 8 - 2 = 54
\]

### (c) \( f(-4) \)

Substitute \( x = -4 \):

\[
f(-4) = 3(-4)^2 + 2(-4) - 2 = 3(16) - 8 - 2 = 48 - 8 - 2 = 38
\]

### (d) \( f(-x) \)

Substitute \( x = -x \):

\[
f(-x) = 3(-x)^2 + 2(-x) - 2 = 3x^2 - 2x - 2
\]

### (e) \( -f(x) \)

Multiply the function by \(-1\):

\[
-f(x) = -(3x^2 + 2x - 2) = -3x^2 - 2x + 2
\]

### (f) \( f(x+2) \)

Substitute \( x = x + 2 \):

\[
f(x+2) = 3(x+2)^2 + 2(x+2) - 2
\]

Simplify:

\[
f(x+2) = 3(x^2 + 4x + 4) + 2x + 4 - 2 = 3x^2 + 12x + 12 + 2x + 2 = 3x^2 + 14x + 14
\]

### (g) \( f(3x) \)

Substitute \( x = 3x \):

\[
f(3x) = 3(3x)^2 + 2(3x) - 2 = 3(9x^2) + 6x - 2 = 27x^2 + 6x - 2
\]

### (h) \( f(x+h) \)

Substitute \( x = x + h \):

\[
f(x+h) = 3(x+h)^2 + 2(x+h) - 2
\]

Simplify:

\[
f(x+h) = 3(x^2 + 2xh + h^2) + 2x + 2h - 2 = 3x^2 + 6xh + 3h^2 + 2x + 2h - 2
\]

Let me know if you would like more details or have any questions.

Here are some related questions:

1. What are the roots of the function \( f(x) = 3x^2 + 2x - 2 \)?
2. How do you find the vertex of the parabola represented by \( f(x) = 3x^2 + 2x - 2 \)?
3. What is the derivative of \( f(x) = 3x^2 + 2x - 2 \)?
4. How do transformations affect the graph of \( f(x) = 3x^2 + 2x - 2 \)?
5. What are the asymptotes of a rational function derived from \( f(x) = 3x^2 + 2x - 2 \)?

**Tip:** Always check for simplification opportunities when substituting into functions, as they can often reduce computation errors and make the problem easier to solve.

Find the following for the function f(x) = 3x^2 + 2x - 2:
(a) f(0)
(b) f(4)
(c) f(-4)
(d) f(-x)
(e) -f(x)
(f) f(x + 2)
(g) f(3x)
(h) f(x + h)

Explore detailed solutions for evaluating the function f(x) = 3x^2 + 2x - 2 at various points and transformations. This guide covers algebraic techniques including substitution, transformations, and simplification. Ideal for students in Grades 9-11.

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