Inverse Functions Test Answer Key with Scoring – Precalculus

Question 1 Domains in Interval Notation (2 points)

1. 2x − 9 ≥ 0 gives x ≥ 9/2: [9/2, ∞).
2. x² − 16 = 0 at x = ±4: (−∞, −4) ∪ (−4, 4) ∪ (4, ∞).

Scoring: 1 point per correct domain.

Question 2 Numerical Compositions (2 points)

1. g(2) = 2, f(2) = 4 + 4 = 8.
2. f(−1) = 1 − 2 = −1, g(−1) = −3 − 4 = −7.

Scoring: 1 point per value.

Question 3 Composite Expressions (2 points)

(f ∘ g)(x) = (x + 3)² − 6(x + 3) = x² + 6x + 9 − 6x − 18 = x² − 9.

(g ∘ f)(x) = x² − 6x + 3 = x² − 6x + 3.

Scoring: 1 point per composite.

Question 4 Domain, Range and Inverse from a Graph (2 points)

1. Domain [−2, ∞), range [−1, ∞).
2. Since f(7) = 2, f⁻¹(2) = 7.

Scoring: 1 point for both intervals; 1 point for the inverse value.

Question 5 Find an Inverse Function (3 points)

Swap: x = 4y + 3y − 2. Then xy − 2x = 4y + 3, so xy − 4y = 2x + 3 and y(x − 4) = 2x + 3.

f⁻¹(x) = 2x + 3x − 4.

Scoring: 1 point for swapping, 1 point for clearing the denominator and collecting y, 1 point for the final formula.

Question 6 Verify Inverses (2 points)

f(g(x)) = ∛((x + 2)³ − 5 + 5) − 2 = (x + 2) − 2 = x.

g(f(x)) = (∛(x + 5))³ − 5 = x + 5 − 5 = x. Both compositions give x.

Scoring: 1 point per composition.

Question 7 Table of an Inverse (2 points)

1. Swap the columns: f⁻¹(7) = −1, f⁻¹(3) = 2, f⁻¹(0) = 5, f⁻¹(−4) = 9.
2. f(f⁻¹(−4)) = −4 and f⁻¹(7) = −1, so the sum is −5.

Scoring: 1 point for the table; 1 point for the sum.

Question 8 Restrict a Parabola, Then Invert (3 points)

1. f(x) = (x − 3)² − 4.
2. Swap: x = (y − 3)² − 4, so y − 3 = √(x + 4) (positive since y ≥ 3): f⁻¹(x) = 3 + √(x + 4).
3. Domain [−4, ∞), range [3, ∞).

Scoring: 1 point per part.

Question 9 Sale Price with Tax (2 points)

1. C(p) = 1.07(0.80p) = 0.856p.
2. 0.856p = 64.20 gives p = 64.20 ÷ 0.856 = $75.

Scoring: 1 point per part.