Quiz on functions operations and composition featuring multiple-choice questions with functions f(x) and g(x) and their compositions.

Let's solve each of the problems step by step.

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Problem 1:

Given:
$f(x) = 2x$, $g(x) = 2x^2 - 1$
Find: $f(g(3))$

Step 1: First compute $g(3)$
$$
g(3) = 2(3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17
$$

Step 2: Now compute $f(g(3)) = f(17)$
$$
f(17) = 2(17) = 34
$$

✔ Answer: A) 34

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Problem 2:

Given:
$f(x) = 2x$, $g(x) = 2x^2 - 1$
Find: $g(f(-1))$

Step 1: Compute $f(-1)$
$$
f(-1) = 2(-1) = -2
$$

Step 2: Compute $g(f(-1)) = g(-2)$
$$
g(-2) = 2(-2)^2 - 1 = 2(4) - 1 = 8 - 1 = 7
$$

✔ Answer: A) 7

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Problem 3:

Given:
$f(x) = 2x$, $g(x) = 2x^2 - 1$
Find: $f(g(x))$

This is a composition: $f(g(x))$

Step: Substitute $g(x)$ into $f(x)$:
$$
f(g(x)) = f(2x^2 - 1) = 2(2x^2 - 1) = 4x^2 - 2
$$

✔ Answer: C) $4x^2 - 2$

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Problem 4:

Given:
$f(x) = 2x$, $g(x) = 2x^2 - 1$
Find: $g(f(x))$

Step: Substitute $f(x)$ into $g(x)$:
$$
g(f(x)) = g(2x) = 2(2x)^2 - 1 = 2(4x^2) - 1 = 8x^2 - 1
$$

✔ Answer: D) $8x^2 - 1$

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Problem 5:

Given:
$f(x) = \frac{1}{x}$, $g(x) = 3x + 2$
Find: $(f \circ g)(2)$

This means $f(g(2))$

Step 1: Compute $g(2)$
$$
g(2) = 3(2) + 2 = 6 + 2 = 8
$$

Step 2: Compute $f(8)$
$$
f(8) = \frac{1}{8}
$$

✔ Answer: A) $\frac{1}{8}$

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Problem 6:

Given:
$f(x) = \frac{1}{x}$, $g(x) = 3x + 2$
Find: $(g \circ g)(-4)$

This means $g(g(-4))$

Step 1: Compute $g(-4)$
$$
g(-4) = 3(-4) + 2 = -12 + 2 = -10
$$

Step 2: Compute $g(-10)$
$$
g(-10) = 3(-10) + 2 = -30 + 2 = -28
$$

✔ Answer: $-28$

But wait — the options are not listed for question 6. However, based on the pattern, we can assume the answer is -28, but since no choices are shown, just report it.

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✔ Final Answers:

1. A) 34
2. A) 7
3. C) $4x^2 - 2$
4. D) $8x^2 - 1$
5. A) $\frac{1}{8}$
6. $-28$

Let me know if you'd like the rest of the 26 questions solved!