Functions and Their Graphs — High School Pre-Calculus practice questions

If f(x) = x² and g(x) = x + 1, find (f ∘ g)(2).

1. 5
2. 4
3. 9 (g(2) = 3, then f(3) = 9)
4. 7
5. 3

Answer: C — 9 (g(2) = 3, then f(3) = 9)

Composition (f∘g)(2) = f(g(2)). First g(2) = 2+1 = 3. Then f(3) = 3² = 9. Order matters: (f∘g)(x) means g first, then f. Compare (g∘f)(2) = g(f(2)) = g(4) = 5.

Convert 60° to radians.

1. π/2
2. π/6
3. π/4
4. π/3 (60° × π/180° = π/3)
5. 2π/3

Answer: D — π/3 (60° × π/180° = π/3)

Convert degrees to radians: multiply by π/180°. 60 · π/180 = π/3. Common angle equivalents: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 360°=2π. Memorize.

Which statement about inverse functions is FALSE?

1. Every function has an inverse function on its natural domain
2. (f⁻¹ ∘ f)(x) = x for all x in f's domain (when f⁻¹ exists)
3. The graphs of f and f⁻¹ are reflections across y = x
4. f⁻¹ exists iff f is one-to-one

Answer: A — Every function has an inverse function on its natural domain

A) FALSE — many functions (e.g., y = x², y = sin x on ℝ) are not one-to-one and have no inverse without domain restriction. This is a classic misconception. B, C, D) All standard true statements about inverses.

If f(x) = 2x + 5 and g(x) = x², find (f ∘ g)(3).

1. 23
2. 19
3. 35
4. 121

Answer: A — 23

A) (f∘g)(3) = f(g(3)) = f(9) = 2(9)+5 = 23. B) Mistakenly computed (g∘f)(3) = g(11) ≠ 19. C) Likely 2(15) + 5 or other arithmetic slip. D) f(3)² = 11² = 121, swapping composition order.

Simplify: sin²(x) + cos²(x).

1. 0
2. sin(2x)
3. 1 (Pythagorean identity)
4. 2
5. cos²(x) - sin²(x)

Answer: C — 1 (Pythagorean identity)

Pythagorean identity: sin²x + cos²x = 1. Foundation of trigonometric identities. Other forms: 1 + tan²x = sec²x; 1 + cot²x = csc²x. Memorize.

What is the inverse of f(x) = (x − 3)/2?

1. f⁻¹(x) = 2x − 3
2. f⁻¹(x) = (x + 3)/2
3. f⁻¹(x) = 2x + 3
4. f⁻¹(x) = 2/(x − 3)

Answer: C — f⁻¹(x) = 2x + 3

C) y = (x − 3)/2 → swap and solve: x = (y − 3)/2 → y = 2x + 3. A) Missed sign on the constant term after isolating y. B) Only swapped without inverting the operation. D) Took the reciprocal of the expression instead of inverting the function.

Find the inverse of f(x) = (2x − 1)/3.

1. f⁻¹(x) = (3x + 1)/2
2. f⁻¹(x) = 3/(2x − 1)
3. f⁻¹(x) = (3x − 1)/2
4. f⁻¹(x) = (2x + 1)/3

Answer: A — f⁻¹(x) = (3x + 1)/2

A) Swap: x = (2y − 1)/3 → 3x = 2y − 1 → y = (3x + 1)/2. B) Confused inverse with reciprocal — a common error. C) Sign error on the constant after isolating y. D) Returned the original function form.

Which function does NOT have an inverse function on its entire natural domain?

1. f(x) = 2x + 1
2. f(x) = x³
3. f(x) = x² (fails the horizontal line test on ℝ)
4. f(x) = eˣ

Answer: C — f(x) = x² (fails the horizontal line test on ℝ)

C) x² maps both x and −x to the same output, failing the horizontal line test on ℝ. We must restrict to x ≥ 0 to invert. A) Linear with nonzero slope: one-to-one. B) Strictly increasing: one-to-one. D) Strictly increasing: one-to-one. Common misconception: 'every function has an inverse' is FALSE.

What is the amplitude of y = 3sin(2x)?

1. 1
2. 3 (amplitude is |a| in y = a·sin(bx); period = 2π/|b|)
3. 2
4. 6
5. 1/2

Answer: B — 3 (amplitude is |a| in y = a·sin(bx); period = 2π/|b|)

Sinusoidal function y = a·sin(bx + c) + d. Amplitude = |a|, Period = 2π/|b|, phase shift = -c/b, vertical shift = d. Here amplitude 3, period π. AP Pre-Calc fundamentals.

What does y = f(x − h) do?

Horizontal shift right by h (left if h < 0).

What is an even function? Give a graphical property.

f(−x) = f(x). Symmetric about the y-axis. Example: x², cos x.

What is function composition (f∘g)(x)?

(f∘g)(x) = f(g(x)) — apply g first, then f.

What does y = f(bx) do (|b| > 1)?

Horizontal compression by factor 1/b. If b < 0, also reflects across the y-axis.