Bluebook SAT Math: Advanced Functions Questions and Answers Flashcards

Let f(x) = x/(x + 2) and g(x) = (x + 2)/(x − 1). For which values of x is f(g(x)) undefined?

Answer: x = 0 and x = 1

f(g(x)) is undefined when (1) g(x) itself is undefined, or (2) g(x) lands on a value that makes f undefined. g(x) is undefined at x = 1. f is undefined when its input equals −2, so set g(x) = −2: (x+2)/(x−1) = −2 → x+2 = −2x+2 → 3x = 0 → x = 0. Thus f(g(x)) is undefined at x = 0 and x = 1.

A function h is defined as h(x) = 2f(3x − 6) + 5, where f is some function. The graph of f has a local maximum at the point (4, 7). What are the coordinates of the corresponding local maximum on the graph of h?

Answer: (10/3, 19)

To find the x-coordinate on h corresponding to x = 4 on f, solve 3x − 6 = 4, giving x = 10/3. The y-coordinate becomes h(10/3) = 2f(4) + 5 = 2(7) + 5 = 19. So the local maximum on h is at (10/3, 19).

If f(x) = √(4x − 3) + 1, what is f⁻¹(5)?

Answer: 19/4

Set f(x) = 5: √(4x − 3) + 1 = 5 → √(4x − 3) = 4 → 4x − 3 = 16 → 4x = 19 → x = 19/4. So f⁻¹(5) = 19/4.

Let p(x) be a polynomial such that p(x + 2) = x³ − 3x + 4 for all real x. What is p(0)?

Answer: 2

To find p(0), we need the input of p to equal 0, so set x + 2 = 0, giving x = −2. Then p(0) = p(−2 + 2) = (−2)³ − 3(−2) + 4 = −8 + 6 + 4 = 2.

A function g satisfies g(g(x)) = x for all real numbers x. If g(3) = 7, what is the value of g(7) + g(3)?

Answer: 10

Since g(g(x)) = x, g is its own inverse. Applying this with x = 3: g(g(3)) = 3, so g(7) = 3. Therefore g(7) + g(3) = 3 + 7 = 10.

A function f defined for all real numbers satisfies f(x + y) = f(x) · f(y) for all real x and y, and f(1) = 3. What is f(0) + f(−1)?

Answer: 4/3

For f(0): set x = y = 0 → f(0) = [f(0)]², so f(0) = 0 or 1. Since f(1) = 3 ≠ 0, f(0) = 0 would force f(everything) = 0 — contradiction. So f(0) = 1. For f(−1): set x = 1, y = −1 → f(0) = f(1)·f(−1) → 1 = 3·f(−1) → f(−1) = 1/3. Therefore f(0) + f(−1) = 1 + 1/3 = 4/3.