Bluebook SAT Test Algebra and Functions 6

Question 1

A function f satisfies f(2x) = 3f(x) − 2 for all real numbers x. If f(1) = 4, what is f(4)?

Accepted Answer

28

Answer

Apply the functional equation step by step. f(2·1) = 3f(1) − 2 = 3(4) − 2 = 10, so f(2) = 10. Then f(2·2) = 3f(2) − 2 = 3(10) − 2 = 28, so f(4) = 28.

Question 2

The function f(x) = (x² − 9) / (x² − x − 6) has a removable discontinuity (hole) at x = a and a vertical asymptote at x = b. What is a − b?

Accepted Answer

5

Answer

Factor numerator and denominator: x² − 9 = (x − 3)(x + 3) and x² − x − 6 = (x − 3)(x + 2). The common factor (x − 3) cancels, creating a hole at x = 3, so a = 3. The remaining denominator gives a vertical asymptote at x = −2, so b = −2. Therefore a − b = 3 − (−2) = 5.

Question 3

The parabola y = x² + 4x + c is tangent to the line y = 2x + 1 (meaning they intersect at exactly one point). What is the value of c?

Accepted Answer

2

Answer

Set the equations equal: x² + 4x + c = 2x + 1, which gives x² + 2x + (c − 1) = 0. For exactly one intersection, the discriminant must equal zero: (2)² − 4(1)(c − 1) = 0 → 4 − 4c + 4 = 0 → 8 = 4c → c = 2. Verify: x² + 2x + 1 = (x + 1)² = 0 has exactly one solution at x = −1. ✓

Question 4

Let f(x) = 3x − 5. If f⁻¹(g(x)) = 2x + 1 for all x, what is g(7)?

Accepted Answer

40

Answer

First find f⁻¹: if y = 3x − 5, then x = (y + 5)/3, so f⁻¹(y) = (y + 5)/3. Substituting into f⁻¹(g(x)) = 2x + 1: (g(x) + 5)/3 = 2x + 1 → g(x) + 5 = 6x + 3 → g(x) = 6x − 2. Therefore g(7) = 6(7) − 2 = 40.

Question 5

Consider f(x) = (3x² − 5x + 2) / (x² − 4). What is the sum of all x-values at which f(x) is either undefined or equal to zero?

Accepted Answer

5/3

Answer

f(x) is undefined when x² − 4 = 0, giving x = 2 and x = −2. f(x) = 0 when the numerator equals zero: 3x² − 5x + 2 = 0 → (3x − 2)(x − 1) = 0 → x = 2/3 and x = 1. Note: neither 2/3 nor 1 make the denominator zero, so both are valid zeros. The sum is 2 + (−2) + 2/3 + 1 = 1 + 2/3 = 5/3.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Algebra and Functions 6