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AP Precalculus study materials
AP Precalculus study resources

AP Precalculus practice questions

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Question 1General Functions (non-analytical)

If f(x) = 3x - 5, what is f(4)?

  1. 2
  2. 12
  3. 17
  4. 7 Correct answer.

Explanation

Substitute x = 4 into the rule: f(4) = 3(4) - 5 = 12 - 5 = 7.

Question 2General Functions (non-analytical)

The function f has domain [-2, 5) and range (0, 8]. Which statement must be true?

  1. The value f(6) is defined
  2. f(-2) is defined Correct answer.
  3. f(5) is defined and equals 8
  4. f(0) must equal 0

Explanation

A square bracket includes the endpoint, so x = -2 is in the domain and f(-2) is defined. The parenthesis at 5 excludes x = 5, and x = 6 lies outside the domain entirely.

Question 3General Functions (non-analytical)

Let f(x) = x^2 + 1 and g(x) = 2x - 3. What is (f ∘ g)(2)?

  1. 2
  2. 1
  3. 5 Correct answer.
  4. 10

Explanation

Composition works inside out: g(2) = 2(2) - 3 = 1, then f(1) = 1^2 + 1 = 2. Wait-recompute carefully: f(1) = 2, so the value is 2.

Question 4General Functions (non-analytical)

A function f is even. Which equation must hold for every x in its domain?

  1. f(0) = 0
  2. f(-x) = -f(x)
  3. f(-x) = f(x) Correct answer.
  4. f(x + 1) = f(x)

Explanation

Even functions are symmetric about the y-axis, which means the outputs at x and -x agree: f(-x) = f(x).

Question 5General Functions (non-analytical)

The graph of y = f(x) is shifted 3 units left and 2 units down. Which equation gives the new graph?

  1. y = f(x + 3) + 2
  2. y = f(x - 3) + 2
  3. y = f(x + 3) - 2 Correct answer.
  4. y = f(x - 3) - 2

Explanation

A left shift of 3 replaces x with x + 3, and a downward shift of 2 subtracts 2 from the output. Combining both gives y = f(x + 3) - 2.

Question 6General Functions (non-analytical)

For the function f(x) = 1/(x - 4), what is the vertical asymptote?

  1. y = 4
  2. x = -4
  3. x = 4 Correct answer.
  4. y = 0

Explanation

A vertical asymptote occurs where the denominator is zero and the numerator is nonzero. Solving x - 4 = 0 gives x = 4.

Question 7General Functions (non-analytical)

A function f is increasing on the interval [1, 6]. If f(2) = 5 and f(5) = 9, which value is possible for f(3)?

  1. 4
  2. 10
  3. 5
  4. 7 Correct answer.

Explanation

Increasing on [1, 6] means f(2) < f(3) < f(5), so 5 < f(3) < 9. Only 7 satisfies this strict inequality.

Question 8General Functions (non-analytical)

If f(x) = 2x + 7, what is f^{-1}(x)?

  1. (x + 7)/2
  2. 1/(2x + 7)
  3. (x - 7)/2 Correct answer.
  4. 2x - 7

Explanation

Set y = 2x + 7 and solve for x: x = (y - 7)/2. Swapping the variable name gives f^{-1}(x) = (x - 7)/2.

Question 9General Functions (non-analytical)

Let f(x) = x^2 - 4x. What is the average rate of change of f on the interval [1, 5]?

  1. 0
  2. 4
  3. 2 Correct answer.
  4. 1

Explanation

The average rate of change is (f(5) - f(1))/(5 - 1). f(5) = 25 - 20 = 5 and f(1) = 1 - 4 = -3, so (5 - (-3))/4 = 8/4 = 2.

Question 10General Functions (non-analytical)

The graph of f is symmetric about the origin. Which statement is consistent with this symmetry?

  1. f(-3) = f(3)
  2. f(-3) = -f(3) Correct answer.
  3. f(-3) = 3
  4. f(3) = 0

Explanation

Origin symmetry means f is odd, so f(-x) = -f(x) for all x in the domain. Applying this at x = 3 gives f(-3) = -f(3).

Question 11General Functions (non-analytical)

Let f(x) = x + 2 and g(x) = x^2. Which expression equals (g ∘ f)(x)?

  1. x^2 + 2
  2. x^2 + 4x + 4 Correct answer.
  3. x^2 + 2x
  4. (x + 2)^3

Explanation

(g ∘ f)(x) = g(f(x)) = g(x + 2) = (x + 2)^2 = x^2 + 4x + 4.

Question 12General Functions (non-analytical)

The function f(x) = x^3 - x has a zero at x = 1. Given that f is odd, which other input must also be a zero?

  1. x = -1 Correct answer.
  2. x = 0
  3. x = 2
  4. x = 1/2

Explanation

For an odd function, if f(1) = 0 then f(-1) = -f(1) = 0. So x = -1 is also a zero by the odd symmetry.

Question 13General Functions (non-analytical)

A function f is defined by f(x) = 3 for x < 0 and f(x) = x + 3 for x ≥ 0. What is f(f(-2))?

  1. 3
  2. 0
  3. 6 Correct answer.
  4. 9

Explanation

First f(-2) = 3 because -2 < 0. Then f(3) = 3 + 3 = 6 because 3 ≥ 0.

Question 14General Functions (non-analytical)

The function f has a removable discontinuity at x = 2 and a jump discontinuity at x = 5. Which statement is consistent with these facts?

  1. f is continuous at x = 2 but not at x = 5
  2. f must be constant near x = 2
  3. f is undefined at x = 2 and x = 5
  4. f is continuous everywhere except possibly at x = 2 and x = 5 Correct answer.

Explanation

Discontinuities are isolated points where continuity fails; elsewhere on its domain f can be continuous. Both x = 2 and x = 5 are points of discontinuity.

Question 15General Functions (non-analytical)

Let h(x) = |x - 3| + 2. What are the vertex and the range of h?

  1. Vertex (3, 2); range (-∞, 2]
  2. Vertex (-3, 2); range [2, ∞)
  3. Vertex (2, 3); range [3, ∞)
  4. Vertex (3, 2); range [2, ∞) Correct answer.

Explanation

The absolute value expression is zero at x = 3, giving the minimum output 2, so the vertex is (3, 2). Since |x - 3| ≥ 0, the outputs are at least 2.

Question 16General Functions (non-analytical)

The graph of y = f(x) is reflected across the x-axis and then stretched vertically by a factor of 2. Which equation describes the result?

  1. y = 2f(x)
  2. y = -f(x)/2
  3. y = f(-2x)
  4. y = -2f(x) Correct answer.

Explanation

Reflecting across the x-axis replaces y with -y, giving -f(x). A vertical stretch by 2 multiplies the output by 2, yielding y = -2f(x).

Question 17General Functions (non-analytical)

Let f(x) = 2x + 1. For what value of x does f(x) = f^{-1}(x)?

  1. x = 1
  2. x = 2
  3. x = -1 Correct answer.
  4. x = 0

Explanation

First f^{-1}(x) = (x - 1)/2. Setting 2x + 1 = (x - 1)/2 gives 4x + 2 = x - 1, so 3x = -3 and x = -1.

Question 18General Functions (non-analytical)

A function f is defined on all real numbers and satisfies f(x + 4) = f(x) for every x. If f(1) = 6, what is f(13)?

  1. 2
  2. 6 Correct answer.
  3. 18
  4. Cannot be determined

Explanation

The condition says f has period 4. Since 13 = 1 + 3(4), applying the periodicity three times gives f(13) = f(1) = 6.

Question 19General Functions (non-analytical)

The function f(x) = (x^2 - 9)/(x - 3) has a hole in its graph. At what point does the hole occur?

  1. (-3, 0)
  2. (0, -3)
  3. (3, 6) Correct answer.
  4. (3, 0)

Explanation

Factor the numerator as (x - 3)(x + 3). Canceling the common factor leaves x + 3 for x ≠ 3, so the hole is at x = 3 with y-value 3 + 3 = 6.

Question 20General Functions (non-analytical)

Let f(x) = x^2 for x ≥ 0 and define g as the inverse of f. What is g(9)?

  1. 3 Correct answer.
  2. 4.5
  3. -3
  4. 81

Explanation

The inverse of f(x) = x^2 on x ≥ 0 is g(x) = √x. Therefore g(9) = 3, the nonnegative root.

Question 21General Functions (non-analytical)

Let f be a function with f(2) = 5 and f(6) = 5, and suppose f is continuous on [2, 6]. Which conclusion is guaranteed by the Intermediate Value Theorem?

  1. f has a local maximum on (2, 6)
  2. f is constant on [2, 6]
  3. f is increasing on [2, 6]
  4. f(c) = 5 for some c in (2, 6) Correct answer.

Explanation

Since f(2) = f(6) = 5, the value 5 is attained at interior points as well; by continuity and the equal endpoint values, there is c in (2, 6) with f(c) = 5.

Question 22General Functions (non-analytical)

The function f is odd and defined for all real numbers. Which statement must be true?

  1. f has no zeros
  2. f is increasing
  3. f(0) = 1
  4. f(0) = 0 Correct answer.

Explanation

For an odd function, f(-x) = -f(x). Setting x = 0 gives f(0) = -f(0), so 2f(0) = 0 and f(0) = 0.

Question 23General Functions (non-analytical)

Let f(x) = 2x + 3 and g(x) = x^2 - 1. For how many values of x does (f ∘ g)(x) = (g ∘ f)(x)?

  1. 2 Correct answer.
  2. Infinitely many
  3. 1
  4. 0

Explanation

(f ∘ g)(x) = 2(x^2 - 1) + 3 = 2x^2 + 1 and (g ∘ f)(x) = (2x + 3)^2 - 1 = 4x^2 + 12x + 8. Setting equal: 2x^2 + 12x + 7 = 0, whose discriminant 144 - 56 = 88 > 0, giving two real solutions.

Question 24General Functions (non-analytical)

A function f has domain [0, 4] and range [-1, 3]. The function g is defined by g(x) = f(x - 2) + 1. What are the domain and range of g?

  1. Domain [0, 4], range [-1, 3]
  2. Domain [-2, 2], range [0, 4]
  3. Domain [2, 6], range [0, 4] Correct answer.
  4. Domain [2, 6], range [-2, 2]

Explanation

For f(x - 2) to be defined, x - 2 must lie in [0, 4], so x is in [2, 6]. Adding 1 shifts every output up by 1, so the range becomes [0, 4].

Question 25General Functions (non-analytical)

The function f satisfies f(x) + f(x + 2) = 4 for all x. If f(0) = 1, what is f(6)?

  1. 2
  2. 1
  3. 3 Correct answer.
  4. 0

Explanation

From f(x + 2) = 4 - f(x): f(2) = 3, f(4) = 1, f(6) = 3. The values alternate between 1 and 3, so f(6) = 3.

Question 26General Functions (non-analytical)

The graph of a function f passes through the point (3, 7). If g(x) = f(x) + 4, which point must lie on the graph of g?

  1. (3, 28)
  2. (7, 7)
  3. (3, 11) Correct answer.
  4. (3, 3)

Explanation

Adding 4 to the output of f shifts every point up by 4 units while keeping the input the same. Since f(3) = 7, we get g(3) = 7 + 4 = 11, so (3, 11) is on the graph of g.

Question 27General Functions (non-analytical)

A function f has domain [-3, 5] and range [2, 9]. What are the domain and range of g(x) = f(x + 1)?

  1. Domain [-4, 4], range [3, 10]
  2. Domain [-3, 5], range [3, 10]
  3. Domain [-4, 4], range [2, 9] Correct answer.
  4. Domain [-2, 6], range [2, 9]

Explanation

The input x + 1 must lie in [-3, 5], so x lies in [-4, 4]. A horizontal shift does not change the set of output values, so the range stays [2, 9].

Question 28General Functions (non-analytical)

Let f(x) = 4x - 1 and g(x) = x/4 + 1. What is (f ∘ g)(x)?

  1. x + 3 Correct answer.
  2. x
  3. 16x - 3
  4. x + 1/4

Explanation

Substituting gives f(g(x)) = 4(x/4 + 1) - 1 = x + 4 - 1 = x + 3. The functions are not inverses because g adds 1 after dividing by 4 rather than adding 1/4 before dividing.

Question 29General Functions (non-analytical)

Let f(x) = 3x + 2. What is f^{-1}(11)?

  1. 11
  2. 35
  3. 1/3
  4. 3 Correct answer.

Explanation

Solve 3x + 2 = 11 to get 3x = 9, so x = 3. Equivalently, f^{-1}(x) = (x - 2)/3 and f^{-1}(11) = 9/3 = 3.

Question 30General Functions (non-analytical)

The function f is one-to-one with f(2) = 5 and f(7) = 3. What is f^{-1}(3) + f^{-1}(5)?

  1. 9 Correct answer.
  2. 3
  3. 10
  4. 8

Explanation

For a one-to-one function, f^{-1} swaps inputs and outputs, so f^{-1}(3) = 7 and f^{-1}(5) = 2. Their sum is 9.

Question 31General Functions (non-analytical)

A function f is even and defined for all real numbers. Its graph passes through (4, -2). Which other point must also lie on the graph of f?

  1. (-4, 2)
  2. (-2, 4)
  3. (4, 2)
  4. (-4, -2) Correct answer.

Explanation

An even function satisfies f(-x) = f(x), so the graph is symmetric about the y-axis. The point (4, -2) reflects to (-4, -2).

Question 32General Functions (non-analytical)

The function f is odd and defined for all real numbers, and its graph passes through (-3, 5). What is f(3) - f(-3)?

  1. -10 Correct answer.
  2. 10
  3. 5
  4. 0

Explanation

For an odd function, f(-x) = -f(x), so f(3) = -f(-3) = -5. Then f(3) - f(-3) = -5 - 5 = -10.

Question 33General Functions (non-analytical)

Let f(x) = x^3 - 4x. The function f is odd. Which statement about the zeros of f is guaranteed by this symmetry?

  1. The zeros of f must sum to 4.
  2. If x = 2 is a zero, then x = -2 is a zero of multiplicity two.
  3. If x = 2 is a zero, then x = -2 is also a zero. Correct answer.
  4. f has no negative zeros.

Explanation

Since f(-x) = -f(x), we have f(-a) = -f(a). If f(a) = 0, then f(-a) = 0 as well, so zeros occur in opposite pairs.

Question 34General Functions (non-analytical)

The function f is continuous on the interval [1, 6], with f(1) = -3 and f(6) = 4. Which value is guaranteed to be assumed by f somewhere on (1, 6)?

  1. 4.5
  2. 7
  3. -5
  4. 0 Correct answer.

Explanation

The Intermediate Value Theorem says a continuous function on a closed interval takes every value between its endpoint values. Since -3 < 0 < 4, the value 0 is attained somewhere in the interval.

Question 35General Functions (non-analytical)

Let f(x) = (x^2 - 1)/(x - 1) for x ≠ 1, and f(1) = 3. Which statement describes f at x = 1?

  1. f has a removable discontinuity at x = 1 because the limit exists but differs from f(1). Correct answer.
  2. f has a jump discontinuity at x = 1 because the left and right limits differ.
  3. f is continuous at x = 1 because f(1) is defined.
  4. f has a vertical asymptote at x = 1.

Explanation

Factoring gives (x - 1)(x + 1)/(x - 1) = x + 1 for x ≠ 1, so the limit as x approaches 1 is 2. Since f(1) = 3 ≠ 2, the discontinuity is removable but the point is misplaced.

Question 36General Functions (non-analytical)

A function f is defined by f(x) = x + 2 for x < 1 and f(x) = x^2 for x ≥ 1. Which statement about f at x = 1 is true?

  1. f has a jump discontinuity at x = 1. Correct answer.
  2. f has a removable discontinuity at x = 1.
  3. f is undefined at x = 1.
  4. f is continuous at x = 1.

Explanation

The left-hand limit is 1 + 2 = 3 while the right-hand value is 1^2 = 1. Because the one-sided limits differ, f has a jump discontinuity at x = 1.

Question 37General Functions (non-analytical)

Let f(x) = 2x - 5. What is the average rate of change of f on the interval [-1, 4]?

  1. 2 Correct answer.
  2. 5
  3. 3
  4. -2

Explanation

For a linear function the average rate of change equals the slope on every interval. Here the slope is 2, and indeed (f(4) - f(-1))/(4 - (-1)) = (3 - (-7))/5 = 2.

Question 38General Functions (non-analytical)

A function f is decreasing on the interval [0, 8]. If f(1) = 10 and f(6) = 4, which value is possible for f(4)?

  1. 6 Correct answer.
  2. 4
  3. 11
  4. 2

Explanation

Because f is decreasing, larger inputs give smaller outputs, so 4 < f(4) < 10 since 1 < 4 < 6. The value 6 lies strictly between 4 and 10 and is therefore possible.

Question 39General Functions (non-analytical)

A function f satisfies f(x + 3) = f(x) for all real x. If f(2) = -1, what is f(17)?

  1. 2
  2. 17
  3. -1 Correct answer.
  4. 3

Explanation

The condition means f is periodic with period 3, so f(x) depends only on x mod 3. Since 17 = 2 + 5·3, we get f(17) = f(2) = -1.

Question 40General Functions (non-analytical)

A function f satisfies f(x) + f(x + 3) = 10 for all x. If f(1) = 4, what is f(7)?

  1. 1
  2. 10
  3. 6
  4. 4 Correct answer.

Explanation

From the relation, f(x + 3) = 10 - f(x). Applying it twice: f(4) = 10 - 4 = 6 and f(7) = 10 - 6 = 4, so the values alternate.

Question 41General Functions (non-analytical)

Let f(x) = |x + 2| - 3. What are the vertex of the graph and the range of f?

  1. Vertex (-2, -3), range [-3, ∞) Correct answer.
  2. Vertex (-2, 3), range [3, ∞)
  3. Vertex (-2, -3), range (-∞, -3]
  4. Vertex (2, -3), range [-3, ∞)

Explanation

The expression |x + 2| is minimized at x = -2, giving f(-2) = -3, so the vertex is (-2, -3). Since an absolute value is never negative, the outputs are at least -3.

Question 42General Functions (non-analytical)

Let f(x) = 3 - |x - 1|. On which interval is f increasing?

  1. (-∞, 1) Correct answer.
  2. (-∞, 3)
  3. (1, ∞)
  4. (3, ∞)

Explanation

The graph is an inverted V with vertex at (1, 3), opening downward. To the left of x = 1 the values rise toward the vertex, so f is increasing on (-∞, 1).

Question 43General Functions (non-analytical)

The graph of y = f(x) contains the point (2, 5). If g(x) = f(-x), which point lies on the graph of g?

  1. (-2, -5)
  2. (5, 2)
  3. (-2, 5) Correct answer.
  4. (2, -5)

Explanation

Replacing x with -x reflects the graph across the y-axis. The point (2, 5) maps to (-2, 5) because g(-2) = f(2) = 5.

Question 44General Functions (non-analytical)

Let f(x) = x^2 for x ≥ 0 and g(x) = x^2 for x ≤ 0. What is the domain of (f + g)(x)?

  1. (-∞, 0]
  2. All real numbers Correct answer.
  3. Only x = 0
  4. [0, ∞)

Explanation

The domain of a sum is the intersection of the domains, but here the sum simplifies because f and g agree on x = 0 and each covers one side. Combined, f + g is defined for every real x, giving domain all real numbers.

Question 45General Functions (non-analytical)

A function f has domain (-∞, 3) ∪ (3, ∞). Which expression could define f?

  1. sqrt(x - 3)
  2. |x - 3|
  3. sqrt(3 - x)
  4. 1/(x - 3) Correct answer.

Explanation

The domain excludes exactly x = 3, which happens when a denominator is zero. The expression 1/(x - 3) is undefined only at x = 3.

Question 46General Functions (non-analytical)

Let f(x) = 2x + 1 and g(x) = (x - 1)/2. Which statement is true?

  1. f and g are inverses, so f(g(x)) = x for all x. Correct answer.
  2. f and g are inverses only when x > 0.
  3. f and g are inverses only at x = 0.
  4. f and g are not inverses because their graphs cross.

Explanation

Composing gives f(g(x)) = 2·((x - 1)/2) + 1 = x - 1 + 1 = x, and similarly g(f(x)) = x. Since both compositions return x for every real x, the functions are inverses.

Question 47General Functions (non-analytical)

A function f is defined on all real numbers, and f(x) = f(x + 5) for every x. If f has exactly 4 zeros in the interval [0, 5), how many zeros does f have in [0, 20)?

  1. 20
  2. 4
  3. 5
  4. 16 Correct answer.

Explanation

The condition makes f periodic with period 5, so the pattern of zeros repeats on each interval of length 5. The interval [0, 20) contains four full periods, giving 4·4 = 16 zeros.

Question 48General Functions (non-analytical)

Let f(x) = x^2 - 2x and g(x) = f(x + 3). What is the vertex of the graph of g?

  1. (-2, -1) Correct answer.
  2. (-2, 8)
  3. (1, -1)
  4. (4, -1)

Explanation

Complete the square: f(x) = (x - 1)^2 - 1, so f has vertex (1, -1). Since g(x) = f(x + 3), the graph shifts left by 3, moving the vertex to (1 - 3, -1) = (-2, -1).

Question 49General Functions (non-analytical)

Let f(x) = x/(x - 2). Which statement about the graph of f is true?

  1. It has a vertical asymptote at x = 2 and no horizontal asymptote.
  2. It has a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. Correct answer.
  3. It has a vertical asymptote at x = 0 and a horizontal asymptote at y = 2.
  4. It has a horizontal asymptote at y = 0 and a hole at x = 2.

Explanation

The denominator is zero at x = 2 while the numerator is not, giving a vertical asymptote there. For large |x| the ratio x/(x - 2) approaches 1, so y = 1 is a horizontal asymptote.

Question 50General Functions (non-analytical)

A function f is continuous and strictly increasing on [0, 10], with f(0) = 1 and f(10) = 21. Which statement must be true?

  1. f has an inverse on [1, 21]. Correct answer.
  2. The range of f is [0, 10].
  3. f(5) = 11.
  4. f has a local maximum in (0, 10).

Explanation

A strictly increasing continuous function is one-to-one, so it has an inverse defined on its range [1, 21]. The Intermediate Value Theorem guarantees every value in that range is attained exactly once.