Evaluating Transformed Functions - Math 2

Given $f(x) = 3x^2 - 2x + 1$ , if $f(a) = f(b)$ where $a \neq b$ , what is $f\left(\frac{a + b}{2}\right)$ ?
Answer: $\frac{2}{3}$ 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.
Given $u(t) = 3t + 1$ and $v(t) = u(t - 2) + u(t + 2)$ , which expression is equivalent to $v(t)$ ?
Answer: $6t + 2$ 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.
Let $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ . If $h(x) = f(g(x))$ , what is the value of $h(-2) + h(1)$ ?
Answer: 22 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.
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 $q(t)$ ?
Answer: $3$ 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.
Let $r(x) = |x - 4|$ and $s(x) = r(2x) - r(x/2)$ . What is the value of $s(6)$ ?
Answer: $7$ 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.
Let $h(x) = |2x - 6|$ . For which value of $k$ is $h(k + 1) = h(k - 1)$ ?
Answer: 3 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.
Given $k(x) = 2^x$ and $\ell(x) = k(x + 2) - 4k(x)$ , which statement about $\ell(x)$ is true?
Answer: $\ell(x) = 0$ for all values of $x$ 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.
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 $h(3)$ ?
Answer: $1$ 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.

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