Question

find the following for the function \\(f(x) = 3x^2 + 3x - 4\\).

(a) \\(f(0)\\)
(b) \\(f(4)\\)
(c) \\(f(-4)\\)
(d) \\(f(-x)\\)
(e) \\(-f(x)\\)
(f) \\(f(x + 3)\\)
(g) \\(f(4x)\\)
(h) \\(f(x + h)\\)

Explanation:

<pre_analysis>
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<reasoning>

Evaluate the function at numerical values

Using the Function Evaluation and Polynomial Evaluation knowledge points:
Substitute the given values into \(f(x) = 3x^2 + 3x - 4\).

• For (a) \(f(0)\):

\[f(0) = 3(0)^2 + 3(0) - 4 = -4\]

• For (b) \(f(4)\):

\[f(4) = 3(4)^2 + 3(4) - 4 = 3(16) + 12 - 4 = 48 + 12 - 4 = 56\]

• For (c) \(f(-4)\):

\[f(-4) = 3(-4)^2 + 3(-4) - 4 = 3(16) - 12 - 4 = 48 - 12 - 4 = 32\]

Find the negative of the function

Using the Function Evaluation knowledge point:
Multiply the entire function expression by \(-1\).

• For (e) \(-f(x)\):

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

Evaluate with algebraic inputs

Using the Function Evaluation knowledge point:
Substitute \(-x\) and \(4x\) into the function.

• For (d) \(f(-x)\):

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

• For (g) \(f(4x)\):

\[f(4x) = 3(4x)^2 + 3(4x) - 4 = 3(16x^2) + 12x - 4 = 48x^2 + 12x - 4\]

Evaluate with binomial inputs

Using the Function Evaluation knowledge point:
Substitute \(x+3\) and \(x+h\) into the function and expand.

• For (f) \(f(x+3)\):

\[

$$\begin{aligned} f(x+3) &= 3(x+3)^2 + 3(x+3) - 4 \\ &= 3(x^2 + 6x + 9) + 3x + 9 - 4 \\ &= 3x^2 + 18x + 27 + 3x + 5 \\ &= 3x^2 + 21x + 32 \end{aligned}$$

\]

• For (h) \(f(x+h)\):

\[

$$\begin{aligned} f(x+h) &= 3(x+h)^2 + 3(x+h) - 4 \\ &= 3(x^2 + 2xh + h^2) + 3x + 3h - 4 \\ &= 3x^2 + 6xh + 3h^2 + 3x + 3h - 4 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

(a) \(f(0) = -4\)

Question 2

(b) \(f(4) = 56\)

Question 3

(c) \(f(-4) = 32\)

Question 4

(d) \(f(-x) = 3x^2 - 3x - 4\)

Question 5

(e) \(-f(x) = -3x^2 - 3x + 4\)

Question 6

(f) \(f(x+3) = 3x^2 + 21x + 32\)

Question 7

(g) \(f(4x) = 48x^2 + 12x - 4\)

Question 8

(h) \(f(x+h) = 3x^2 + 6xh + 3h^2 + 3x + 3h - 4\)
</answer>

<post_analysis>
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"question_type": "Multi-part",
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"Mathematics",
"Algebra",
"Function Evaluation"
]
}
</post_analysis>

Answer:

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}
</pre_analysis>

<reasoning>

Evaluate the function at numerical values

Using the Function Evaluation and Polynomial Evaluation knowledge points:
Substitute the given values into \(f(x) = 3x^2 + 3x - 4\).

• For (a) \(f(0)\):

\[f(0) = 3(0)^2 + 3(0) - 4 = -4\]

• For (b) \(f(4)\):

\[f(4) = 3(4)^2 + 3(4) - 4 = 3(16) + 12 - 4 = 48 + 12 - 4 = 56\]

• For (c) \(f(-4)\):

\[f(-4) = 3(-4)^2 + 3(-4) - 4 = 3(16) - 12 - 4 = 48 - 12 - 4 = 32\]

Find the negative of the function

Using the Function Evaluation knowledge point:
Multiply the entire function expression by \(-1\).

• For (e) \(-f(x)\):

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

Evaluate with algebraic inputs

Using the Function Evaluation knowledge point:
Substitute \(-x\) and \(4x\) into the function.

• For (d) \(f(-x)\):

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

• For (g) \(f(4x)\):

\[f(4x) = 3(4x)^2 + 3(4x) - 4 = 3(16x^2) + 12x - 4 = 48x^2 + 12x - 4\]

Evaluate with binomial inputs

Using the Function Evaluation knowledge point:
Substitute \(x+3\) and \(x+h\) into the function and expand.

• For (f) \(f(x+3)\):

\[

$$\begin{aligned} f(x+3) &= 3(x+3)^2 + 3(x+3) - 4 \\ &= 3(x^2 + 6x + 9) + 3x + 9 - 4 \\ &= 3x^2 + 18x + 27 + 3x + 5 \\ &= 3x^2 + 21x + 32 \end{aligned}$$

\]

• For (h) \(f(x+h)\):

\[

$$\begin{aligned} f(x+h) &= 3(x+h)^2 + 3(x+h) - 4 \\ &= 3(x^2 + 2xh + h^2) + 3x + 3h - 4 \\ &= 3x^2 + 6xh + 3h^2 + 3x + 3h - 4 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

(a) \(f(0) = -4\)

Question 2

(b) \(f(4) = 56\)

Question 3

(c) \(f(-4) = 32\)

Question 4

(d) \(f(-x) = 3x^2 - 3x - 4\)

Question 5

(e) \(-f(x) = -3x^2 - 3x + 4\)

Question 6

(f) \(f(x+3) = 3x^2 + 21x + 32\)

Question 7

(g) \(f(4x) = 48x^2 + 12x - 4\)

Question 8

(h) \(f(x+h) = 3x^2 + 6xh + 3h^2 + 3x + 3h - 4\)
</answer>

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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Evaluation"
]
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</post_analysis>