Math 2 Quiz: Evaluating Transformed Functions

8 questions · exam conditions

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Evaluating Transformed FunctionsQuestion 1 of 8

Given $k(x) = 2^x$ and $\ell(x) = k(x + 2) - 4k(x)$ , which statement about $\ell(x)$ is true?

\ell(x)

can be positive, negative, or zero depending on

x

\ell(x) > 0

for all values of

x

\ell(x) < 0

for all values of

x

\ell(x) = 0

for all values of

x

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Math 2 Quiz

Math 2 Quiz: Evaluating Transformed Functions

Practice Evaluating Transformed Functions in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Evaluating Transformed Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Given $k(x) = 2^x$ and $\ell(x) = k(x + 2) - 4k(x)$ , which statement about $\ell(x)$ is true?

$\ell(x)$ can be positive, negative, or zero depending on $x$
$\ell(x) > 0$ for all values of $x$
$\ell(x) < 0$ for all values of $x$
$\ell(x) = 0$ for all values of $x$ (correct answer)

Explanation: This problem tests your ability to manipulate exponential functions and recognize when expressions simplify to constants. When you see composite functions involving exponentials, always substitute and simplify systematically. Let's find

\ell(x)

by substituting the given functions. Since

k(x) = 2^x

, we have:

$k(x + 2) = 2^{x+2}$
$4k(x) = 4 \cdot 2^x$

Therefore:

\ell(x) = k(x + 2) - 4k(x) = 2^{x+2} - 4 \cdot 2^x

Now simplify

2^{x+2}

using exponent rules:

2^{x+2} = 2^x \cdot 2^2 = 2^x \cdot 4 = 4 \cdot 2^x

Substituting back:

\ell(x) = 4 \cdot 2^x - 4 \cdot 2^x = 0

Since

\ell(x) = 0

regardless of the value of

x

, answer D is correct. A is wrong because

\ell(x)

doesn't vary with

x

— it's always zero. B is incorrect since

\ell(x)

is never positive; it equals zero. C is wrong because

\ell(x)

is never negative; again, it equals zero. Key strategy: When working with exponential function compositions, always use exponent rules to rewrite terms in a common form. Here, recognizing that

2^{x+2} = 4 \cdot 2^x

was crucial. This type of problem often appears when testing whether students can spot when complex-looking expressions actually simplify to constants.

Question 2

Let $r(x) = |x - 4|$ and $s(x) = r(2x) - r(x/2)$ . What is the value of $s(6)$ ?

$5$
$7$ (correct answer)
$9$
$11$

Explanation: To find s(6), substitute x = 6 into s(x) = r(2x) - r(x/2). First, calculate r(2·6) = r(12) = |12 - 4| = |8| = 8. Then calculate r(6/2) = r(3) = |3 - 4| = |-1| = 1. Therefore, s(6) = r(12) - r(3) = 8 - 1 = 7. Choice A results from calculating |6 - 4| + 1 = 3 instead of the correct transformation. Choice B is the correct answer: 8 - 1 = 7. Choice C might come from adding instead of subtracting: 8 + 1 = 9. Choice D could result from miscalculating one of the absolute values or using wrong inputs.

Question 3

Let $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ . If $h(x) = f(g(x))$ , what is the value of $h(-2) + h(1)$ ?

28
22 (correct answer)
16
10

Explanation: First, find h(x) = f(g(x)). Since g(x) = x² + 1, we have h(x) = f(x² + 1) = 3(x² + 1) - 2 = 3x² + 3 - 2 = 3x² + 1. Then h(-2) = 3(-2)² + 1 = 3(4) + 1 = 13, and h(1) = 3(1)² + 1 = 3(1) + 1 = 4. Therefore, h(-2) + h(1) = 13 + 4 = 22. Choice A results from incorrectly computing h(x) = 3x² + 3. Choice C results from finding h(2) instead of h(-2). Choice D results from computing f(-2) + f(1) instead of the composition.

Question 4

Let $g(x) = \frac{x^2}{x^2 + 1}$ and $h(x) = g(x) + g\left(\frac{1}{x}\right)$ . What is the value of $h(3)$ ?

$\frac{8}{9}$
$\frac{9}{10}$
$1$ (correct answer)
$\frac{10}{9}$

Explanation: To find h(3), substitute x = 3 into h(x) = g(x) + g(1/x). First, calculate g(3) = 3²/(3² + 1) = 9/(9 + 1) = 9/10. Then calculate g(1/3) = (1/3)²/((1/3)² + 1) = (1/9)/(1/9 + 1) = (1/9)/(1/9 + 9/9) = (1/9)/(10/9) = (1/9) × (9/10) = 1/10. Therefore, h(3) = g(3) + g(1/3) = 9/10 + 1/10 = 10/10 = 1. This demonstrates a key property: for this function, g(x) + g(1/x) = 1 for any positive x. Choice A might result from calculation errors in g(1/3). Choice B is just g(3) without adding g(1/3). Choice C is correct: the sum equals 1. Choice D would result from incorrectly inverting the final fraction.

Question 5

Given $u(t) = 3t + 1$ and $v(t) = u(t - 2) + u(t + 2)$ , which expression is equivalent to $v(t)$ ?

$6t + 2$ (correct answer)
$6t - 2$
$6t + 8$
$3t + 5$

Explanation: To find the expression for v(t), substitute into v(t) = u(t - 2) + u(t + 2). First, calculate u(t - 2) = 3(t - 2) + 1 = 3t - 6 + 1 = 3t - 5. Then calculate u(t + 2) = 3(t + 2) + 1 = 3t + 6 + 1 = 3t + 7. Therefore, v(t) = (3t - 5) + (3t + 7) = 6t + 2. Choice A is correct: 6t + 2. Choice B might result from sign errors in combining terms. Choice C could come from miscalculating the constant terms. Choice D represents only one transformation instead of the sum of both.

Question 6

If $p(t) = t^3 - 6t^2 + 11t - 6$ and $q(t) = p(t + 1) - p(t)$ , what is the coefficient of $t^2$ in the simplified form of $q(t)$ ?

$1$
$3$ (correct answer)
$6$
$9$

Explanation: To find the coefficient of t² in q(t) = p(t + 1) - p(t), we need to expand p(t + 1). First, p(t + 1) = (t + 1)³ - 6(t + 1)² + 11(t + 1) - 6. Expanding: (t + 1)³ = t³ + 3t² + 3t + 1, (t + 1)² = t² + 2t + 1, and (t + 1) = t + 1. So p(t + 1) = (t³ + 3t² + 3t + 1) - 6(t² + 2t + 1) + 11(t + 1) - 6 = t³ + 3t² + 3t + 1 - 6t² - 12t - 6 + 11t + 11 - 6 = t³ + (3 - 6)t² + (3 - 12 + 11)t + (1 - 6 + 11 - 6) = t³ - 3t² + 2t + 0 = t³ - 3t² + 2t. Therefore, q(t) = p(t + 1) - p(t) = (t³ - 3t² + 2t) - (t³ - 6t² + 11t - 6) = t³ - 3t² + 2t - t³ + 6t² - 11t + 6 = (-3 + 6)t² + (2 - 11)t + 6 = 3t² - 9t + 6. The coefficient of t² is 3. Choice A might result from errors in expansion. Choice B is correct: the coefficient is 3. Choice C could come from forgetting to subtract terms. Choice D might result from calculation errors in combining like terms.

Question 7

Let $h(x) = |2x - 6|$ . For which value of $k$ is $h(k + 1) = h(k - 1)$ ?

1
2
3 (correct answer)
4

Explanation: We need |2(k + 1) - 6| = |2(k - 1) - 6|, which simplifies to |2k + 2 - 6| = |2k - 2 - 6|, or |2k - 4| = |2k - 8|. For absolute values to be equal, either 2k - 4 = 2k - 8 (impossible since -4 ≠ -8) or 2k - 4 = -(2k - 8) = -2k + 8. Solving 2k - 4 = -2k + 8 gives 4k = 12, so k = 3. Verification: h(4) = |2(4) - 6| = |2| = 2, and h(2) = |2(2) - 6| = |-2| = 2. Choice A gives h(2) = 2 and h(0) = 6. Choice B gives h(3) = 0 and h(1) = 4. Choice D gives h(5) = 4 and h(3) = 0.

Question 8

Given $f(x) = 3x^2 - 2x + 1$ , if $f(a) = f(b)$ where $a \neq b$ , what is $f\left(\frac{a + b}{2}\right)$ ?

$\frac{1}{4}$
$\frac{2}{3}$ (correct answer)
$\frac{3}{4}$
1

Explanation: Since f(a) = f(b), we have 3a² - 2a + 1 = 3b² - 2b + 1. This simplifies to 3a² - 2a = 3b² - 2b, or 3(a² - b²) = 2(a - b). Since a ≠ b, we can divide by (a - b) to get 3(a + b) = 2, so a + b = 2/3. The axis of symmetry of the parabola f(x) = 3x² - 2x + 1 is at x = -(-2)/(2·3) = 1/3. Since f(a) = f(b) with a ≠ b, points a and b are equidistant from the axis of symmetry, so (a + b)/2 = 1/3. Therefore, f((a + b)/2) = f(1/3) = 3(1/3)² - 2(1/3) + 1 = 3(1/9) - 2/3 + 1 = 1/3 - 2/3 + 1 = 2/3.