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CLEP College Mathematics study materials
CLEP College Mathematics study resources

CLEP College Mathematics practice questions

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Question 1Algebra and Functions

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

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

Explanation

Substitute x = 4 into the rule: f(4) = 3(4) - 5 = 12 - 5 = 7. Function notation means the value inside the parentheses replaces every x in the formula.

Question 2Algebra and Functions

What is the domain of the function g(x) = 1/(x - 6)?

  1. All real numbers except x = 0
  2. All real numbers greater than 6
  3. All real numbers
  4. All real numbers except x = 6 Correct answer.

Explanation

A rational function is undefined wherever its denominator equals zero. Setting x - 6 = 0 gives x = 6, so every real number except 6 is allowed.

Question 3Algebra and Functions

Which equation represents a line with slope -2 passing through the point (3, 1)?

  1. y = -2x + 1
  2. y = -2x - 7
  3. y = 3x - 2
  4. y = -2x + 7 Correct answer.

Explanation

Using point-slope form, y - 1 = -2(x - 3), which expands to y = -2x + 6 + 1 = -2x + 7. Checking the point: -2(3) + 7 = 1.

Question 4Algebra and Functions

Simplify (2x^3)(4x^5).

  1. 6x^8
  2. 8x^8 Correct answer.
  3. 6x^15
  4. 8x^15

Explanation

Multiply the coefficients: 2 times 4 equals 8. When multiplying powers with the same base, add the exponents: x^3 times x^5 equals x^8, giving 8x^8.

Question 5Algebra and Functions

The graph of y = x^2 - 4 is shifted 3 units to the right. Which equation gives the new graph?

  1. y = x^2 - 1
  2. y = (x - 3)^2 - 4 Correct answer.
  3. y = (x + 3)^2 - 4
  4. y = x^2 - 7

Explanation

A horizontal shift right by h units replaces x with (x - h). Shifting 3 units right therefore gives y = (x - 3)^2 - 4.

Question 6Algebra and Functions

Solve the equation 5x - 7 = 2x + 11.

  1. x = 6 Correct answer.
  2. x = 18
  3. x = 4
  4. x = 2

Explanation

Subtract 2x from both sides to get 3x - 7 = 11, then add 7 to get 3x = 18, so x = 6. Substituting back gives 23 on both sides.

Question 7Algebra and Functions

If f(x) = x^2 + 1 and g(x) = 2x, what is (f + g)(3)?

  1. 16 Correct answer.
  2. 10
  3. 19
  4. 13

Explanation

Compute each function value: f(3) = 9 + 1 = 10 and g(3) = 6. The sum of the functions at 3 is 10 + 6 = 16.

Question 8Algebra and Functions

Which statement describes the range of the function f(x) = x^2 + 2?

  1. All real numbers y with y greater than 0
  2. All real numbers
  3. All real numbers y with y less than or equal to 2
  4. All real numbers y with y greater than or equal to 2 Correct answer.

Explanation

Since x^2 is never negative, its smallest value is 0, occurring at x = 0. Adding 2 makes the smallest output 2, and outputs grow without bound, so the range is y greater than or equal to 2.

Question 9Algebra and Functions

Find the slope of the line passing through the points (2, 5) and (6, 13).

  1. 4
  2. 8
  3. 2 Correct answer.
  4. 1/2

Explanation

Slope equals the change in y divided by the change in x: (13 - 5)/(6 - 2) = 8/4 = 2.

Question 10Algebra and Functions

What is the solution set of the inequality -3x + 4 < 13?

  1. x > -3 Correct answer.
  2. x > 3
  3. x < -3
  4. x < 3

Explanation

Subtract 4 to get -3x < 9, then divide by -3. Dividing by a negative number reverses the inequality, giving x > -3.

Question 11Algebra and Functions

The function f(x) = 2x + 1 models the cost in dollars of producing x items plus a fixed fee. What does the slope 2 represent?

  1. The cost increases by 2 dollars for each additional item Correct answer.
  2. The average cost per item at every production level
  3. The total cost when no items are produced
  4. The number of items produced

Explanation

In a linear model the slope measures the rate of change of the output per unit change in the input. Here each extra item raises the cost by 2 dollars, while the intercept 1 is the fixed fee.

Question 12Algebra and Functions

If f(x) = 4x - 3, what is f^{-1}(9)?

  1. 39
  2. 2
  3. 3 Correct answer.
  4. 9/4

Explanation

Set 4x - 3 = 9 and solve: 4x = 12, so x = 3. The inverse function returns the input that produces the output 9.

Question 13Algebra and Functions

Which expression is the complete factorization of x^2 - 5x - 24?

  1. (x - 6)(x + 4)
  2. (x - 12)(x + 2)
  3. (x - 8)(x + 3) Correct answer.
  4. (x + 8)(x - 3)

Explanation

We need two numbers whose product is -24 and whose sum is -5; those numbers are -8 and 3. Expanding (x - 8)(x + 3) gives x^2 - 5x - 24, confirming the factors.

Question 14Algebra and Functions

The quadratic function h(t) = -16t^2 + 64t gives the height in feet of an object t seconds after launch. At what time does the object reach its maximum height?

  1. 1 second
  2. 4 seconds
  3. 2 seconds Correct answer.
  4. 64 seconds

Explanation

For a parabola y = at^2 + bt + c, the vertex occurs at t = -b/(2a). Here t = -64/(2 times -16) = -64/-32 = 2, so the maximum height is reached at 2 seconds.

Question 15Algebra and Functions

What is the solution set of |2x - 6| = 10?

  1. x = 8 or x = -2 Correct answer.
  2. x = 2 or x = -8
  3. x = 8 only
  4. x = 16 or x = -4

Explanation

An absolute value equation splits into two cases: 2x - 6 = 10 gives x = 8, and 2x - 6 = -10 gives x = -2. Both values check in the original equation.

Question 16Algebra and Functions

If log base 2 of 8 equals 3, which exponential statement is equivalent?

  1. 2^8 = 3
  2. 8^2 = 3
  3. 2^3 = 8 Correct answer.
  4. 3^2 = 8

Explanation

The definition of a logarithm states that log base b of y equals x exactly when b^x = y. Applying this with b = 2, y = 8, and x = 3 gives 2^3 = 8.

Question 17Algebra and Functions

The function f(x) = 3(2)^x models a population. By what factor does the population change when x increases by 4?

  1. 4
  2. 16 Correct answer.
  3. 8
  4. 12

Explanation

Increasing x by 4 multiplies the function by 2^4, because 3(2)^(x+4) = 3(2)^x times 16. The factor is therefore 16, independent of the starting value of x.

Question 18Algebra and Functions

What are the x-intercepts of the graph of y = x^2 - 7x + 12?

  1. x = -3 and x = -4
  2. x = 3 and x = 4 Correct answer.
  3. x = 2 and x = 6
  4. x = 12 and x = 1

Explanation

Set y = 0 and factor: x^2 - 7x + 12 = (x - 3)(x - 4). The x-intercepts occur where each factor is zero, at x = 3 and x = 4.

Question 19Algebra and Functions

If f(x) = x + 2 and g(x) = 3x, what is (f composed with g)(x), meaning f(g(x))?

  1. 3x^2 + 2
  2. 3x + 2 Correct answer.
  3. x + 6
  4. 3x + 6

Explanation

In a composition f(g(x)), the output of g becomes the input of f. Substituting g(x) = 3x into f gives f(3x) = 3x + 2.

Question 20Algebra and Functions

The system 2x + y = 7 and x - y = 2 is solved. What is the value of x?

  1. 9
  2. 1
  3. 3 Correct answer.
  4. 5

Explanation

Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x - y = 2 gives y = 1, and both equations check.

Question 21Algebra and Functions

The function f is defined by f(x) = (x^2 - 9)/(x - 3) for x not equal to 3. Which value should be assigned to f(3) so that the extended function is continuous at x = 3?

  1. 6 Correct answer.
  2. Undefined, because the original formula is undefined there
  3. 3
  4. 0

Explanation

Factor the numerator as (x - 3)(x + 3), so the expression simplifies to x + 3 for x not equal to 3. The limit as x approaches 3 is 6, so defining f(3) = 6 removes the removable discontinuity.

Question 22Algebra and Functions

If f(x) = x^3 + 2x, which statement about f is true?

  1. f is an odd function because f(-x) = -f(x) Correct answer.
  2. f is neither even nor odd because of the constant term
  3. f is an even function because f(-x) = f(x)
  4. f is even because its graph has a y-intercept at 0

Explanation

Compute f(-x) = (-x)^3 + 2(-x) = -x^3 - 2x = -(x^3 + 2x) = -f(x). Since f(-x) equals the negative of f(x), the function is odd and its graph is symmetric about the origin.

Question 23Algebra and Functions

The polynomial p(x) = x^3 - 4x^2 + x + 6 has x = 2 as a zero. What are all the zeros of p?

  1. x = 2, x = 3, and x = 1
  2. x = 2, x = 3, and x = -1 Correct answer.
  3. x = 2 only
  4. x = 2, x = -3, and x = -1

Explanation

Dividing p(x) by (x - 2) leaves the quadratic x^2 - 2x - 3, which factors as (x - 3)(x + 1). The complete factorization is (x - 2)(x - 3)(x + 1), so the zeros are 2, 3, and -1.

Question 24Algebra and Functions

For which values of x is the expression (x - 1)/(x^2 - 4) undefined, and what happens to the expression as x approaches 2 from the right?

  1. Undefined at x = 2 and x = -2; it grows without bound as x approaches 2 from the right Correct answer.
  2. Undefined at x = 2 and x = -2; it approaches 1/4 as x approaches 2
  3. Undefined at x = 1 only; it approaches 0 as x approaches 2
  4. Undefined at x = 0; it approaches infinity as x approaches 2

Explanation

The denominator factors as (x - 2)(x + 2), so the expression is undefined at x = 2 and x = -2. As x approaches 2 from the right, the numerator approaches 1 while the denominator approaches 0 through positive values, so the expression increases without bound.

Question 25Algebra and Functions

The function f(x) = 2x + 1 is applied, and then its inverse is applied to the result. If the starting input is 5, what is the final output?

  1. 5 Correct answer.
  2. 1/11
  3. 23
  4. 11

Explanation

Applying a function and then its inverse returns the original input, since f^{-1}(f(x)) = x for every x in the domain. The intermediate value f(5) = 11, and f^{-1}(11) = (11 - 1)/2 = 5.

Question 26Algebra and Functions

A line passes through the points (-3, 4) and (5, -2). What is the slope of the line?

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

Explanation

The slope m is the change in y divided by the change in x: m = (-2 - 4) / (5 - (-3)) = -6 / 8 = -3/4. This negative value indicates the line falls from left to right. The slope formula is a fundamental concept in algebra and functions.

Question 27Algebra and Functions

Which set correctly describes the domain of the function f(x) = 1/(x - 2)?

  1. All real numbers except x = -2
  2. All real numbers
  3. All real numbers except x = 2 Correct answer.
  4. All real numbers greater than 2

Explanation

The function is undefined when the denominator equals zero, which occurs at x = 2. Therefore, the domain is all real numbers except 2. This is a standard domain restriction for rational functions.

Question 28Algebra and Functions

The function f is defined by f(x) = 2x + 3. What is the value of f^{-1}(7)?

  1. 11
  2. 2 Correct answer.
  3. 17
  4. 5

Explanation

To find f^{-1}(7), solve the equation 2x + 3 = 7. Subtracting 3 gives 2x = 4, so x = 2. Thus f^{-1}(7) = 2. This process reverses the function's operation.

Question 29Algebra and Functions

The graph of y = (x - 3)^2 + 4 is a parabola. What are the coordinates of its vertex?

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

Explanation

The equation is in vertex form y = a(x - h)^2 + k, where the vertex is (h, k). Here h = 3 and k = 4, so the vertex is (3, 4). The squared term is zero when x = 3, giving the minimum y-value of 4.

Question 30Algebra and Functions

The system of equations 3x + 2y = 12 and x - y = 1 is solved. What is the value of y?

  1. 9/5 Correct answer.
  2. 3
  3. 3/5
  4. 1

Explanation

From the second equation, x = y + 1. Substituting into the first gives 3(y + 1) + 2y = 12, so 5y + 3 = 12, leading to 5y = 9 and y = 9/5. This is the unique solution for y.

Question 31Algebra and Functions

What is the solution set of the inequality |x - 4| ≥ 2?

  1. x ≤ -2 or x ≥ 6
  2. x ≤ 2 or x ≥ 6 Correct answer.
  3. x ≤ 2 or x ≥ -6
  4. 2 ≤ x ≤ 6

Explanation

The inequality |x - 4| ≥ 2 means the distance from x to 4 is at least 2. This gives x - 4 ≤ -2 or x - 4 ≥ 2, which simplifies to x ≤ 2 or x ≥ 6. The solution is the union of two rays.

Question 32Algebra and Functions

The function f(x) = 5(0.8)^x models the value of a car after x years. What does the base 0.8 represent?

  1. The car loses 20% of its value each year Correct answer.
  2. The car gains 20% of its value each year
  3. The car loses 80% of its value each year
  4. The car's value remains constant each year

Explanation

In an exponential model f(x) = a(b)^x, the base b is the growth or decay factor. A base of 0.8 means the value is multiplied by 0.8 each year, which is a 20% decrease. Thus the car depreciates by 20% annually.

Question 33Algebra and Functions

The function f(x) = x^2 - 4x + 3 is graphed. What is the equation of its axis of symmetry?

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

Explanation

For a quadratic in standard form f(x) = ax^2 + bx + c, the axis of symmetry is x = -b/(2a). Here a = 1 and b = -4, so x = -(-4)/(2*1) = 2. The axis is the vertical line x = 2.

Question 34Algebra and Functions

If f(x) = 2x + 1 and g(x) = x^2, what is (g ∘ f)(x)?

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

Explanation

The composition (g ∘ f)(x) means g(f(x)). Substitute f(x) into g: g(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1. This expands using the binomial square.

Question 35Algebra and Functions

What is the remainder when the polynomial 2x^3 - 3x^2 + 4x - 5 is divided by x - 1?

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

Explanation

By the Remainder Theorem, the remainder is the value of the polynomial at x = 1. Evaluate: 2(1)^3 - 3(1)^2 + 4(1) - 5 = 2 - 3 + 4 - 5 = -2. Thus the remainder is -2.

Question 36Algebra and Functions

The function f(x) = 3x - 2 is one-to-one. Which expression gives f^{-1}(x)?

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

Explanation

To find the inverse, set y = 3x - 2 and solve for x: y + 2 = 3x, so x = (y + 2)/3. Replacing y with x gives f^{-1}(x) = (x + 2)/3. This correctly undoes the operations of f.

Question 37Algebra and Functions

The expression (x^2 - 9)/(x^2 - x - 6) is simplified. For which values of x is the simplified expression undefined?

  1. x = 3 and x = -3
  2. x = -2 and x = 3 Correct answer.
  3. x = 3 only
  4. x = -2 only

Explanation

Factor numerator and denominator: (x - 3)(x + 3) / [(x - 3)(x + 2)]. Cancel the common factor (x - 3), leaving (x + 3)/(x + 2). The simplified expression is undefined when its denominator is zero, at x = -2. However, the original expression is also undefined at x = 3, which is a removable discontinuity. Thus both x = -2 and x = 3 make the original expression undefined.

Question 38Algebra and Functions

The function f(x) = x^3 - 3x^2 + 4 is analyzed. Which statement about its graph is true?

  1. It has no local extrema
  2. It has a local maximum at x = 0 and a local minimum at x = 2 Correct answer.
  3. It has a local maximum at x = 1 and a local minimum at x = 3
  4. It has a local minimum at x = 0 and a local maximum at x = 2

Explanation

Find the derivative: f'(x) = 3x^2 - 6x = 3x(x - 2). Critical points are x = 0 and x = 2. The second derivative f''(x) = 6x - 6 gives f''(0) = -6 (concave down, local maximum) and f''(2) = 6 (concave up, local minimum). Thus the graph has a local maximum at x = 0 and a local minimum at x = 2.

Question 39Algebra and Functions

The function f(x) = log_2(x + 3) is defined. What is the domain of f?

  1. x > -3 Correct answer.
  2. All real numbers
  3. x ≥ -3
  4. x > 3

Explanation

The argument of a logarithm must be positive, so x + 3 > 0, which gives x > -3. The domain is all real numbers greater than -3. The logarithm is not defined at x = -3.

Question 40Algebra and Functions

The function f(x) = (x^2 - 1)/(x - 1) is considered. Which statement correctly describes the behavior of f near x = 1?

  1. f is continuous at x = 1 and f(1) = 2
  2. f has a jump discontinuity at x = 1
  3. f has a removable discontinuity at x = 1, and the limit as x approaches 1 is 2 Correct answer.
  4. f has a vertical asymptote at x = 1

Explanation

Factor the numerator: (x - 1)(x + 1)/(x - 1). For x ≠ 1, this simplifies to x + 1. As x approaches 1, the limit is 2. The function is undefined at x = 1, so there is a removable discontinuity (a hole) at that point.

Question 41Counting and Probability

A café offers 4 choices of bread and 5 choices of filling for a sandwich. If every sandwich uses exactly one bread and exactly one filling, how many different sandwiches can be made?

  1. 25
  2. 9
  3. 20 Correct answer.
  4. 16

Explanation

The multiplication principle applies because the bread and filling are chosen independently: multiply the number of options for each stage. Thus 4 × 5 = 20 distinct sandwiches.

Question 42Counting and Probability

How many ways can the letters of the word LEVEL be arranged if all five letters are used and identical letters are treated as indistinguishable?

  1. 30
  2. 120
  3. 20 Correct answer.
  4. 60

Explanation

There are 5 letters with L repeated twice and E repeated twice, so the number of distinct arrangements is 5!/(2!·2!) = 120/4. This gives 30? No - compute carefully: 120 divided by 4 equals 30. Rechecking: 5! = 120, 2!·2! = 4, so the count is 30.

Question 43Counting and Probability

A bag holds 7 red marbles and 3 blue marbles. One marble is drawn at random. What is the probability that it is blue?

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

Explanation

Probability equals favorable outcomes divided by total outcomes. There are 3 blue marbles out of 10 total, so P(blue) = 3/10.

Question 44Counting and Probability

From a group of 8 students, a committee of 3 is to be selected. The order of selection does not matter. How many different committees are possible?

  1. 336
  2. 24
  3. 512
  4. 56 Correct answer.

Explanation

Because order does not matter, this is a combination: C(8,3) = 8!/(3!·5!) = (8·7·6)/(3·2·1) = 56.

Question 45Counting and Probability

Two fair six-sided dice are rolled. What is the probability that the sum of the two dice is 7?

  1. 1/9
  2. 1/7
  3. 7/36
  4. 1/6 Correct answer.

Explanation

There are 36 equally likely ordered outcomes. The sum 7 occurs in 6 of them: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Thus P = 6/36 = 1/6.

Question 46Counting and Probability

A fair coin is flipped 4 times. What is the probability of getting exactly 2 heads?

  1. 5/16
  2. 3/8 Correct answer.
  3. 1/4
  4. 1/2

Explanation

There are 2^4 = 16 equally likely sequences. The number with exactly 2 heads is C(4,2) = 6, so the probability is 6/16 = 3/8.

Question 47Counting and Probability

A password consists of 3 letters chosen from the 26-letter alphabet, with letters allowed to repeat, followed by 2 digits chosen from 0 through 9, with digits allowed to repeat. How many such passwords exist?

  1. 26^3 + 10^2
  2. 26^2 · 10^3
  3. 26^3 · 10^2 Correct answer.
  4. 26·25·24·10·9

Explanation

Each of the 3 letter positions has 26 choices and each of the 2 digit positions has 10 choices, all independent. Multiplying gives 26^3 · 10^2 passwords.

Question 48Counting and Probability

In a class of 30 students, 18 play a sport, 12 play a musical instrument, and 6 do both. If one student is chosen at random, what is the probability that the student plays a sport or plays an instrument?

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

Explanation

By the addition rule, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 18/30 + 12/30 − 6/30 = 24/30 = 4/5.

Question 49Counting and Probability

A box contains 5 green and 4 yellow balls. Two balls are drawn without replacement. What is the probability that both are green?

  1. 5/18 Correct answer.
  2. 1/2
  3. 5/9
  4. 25/81

Explanation

The probability the first is green is 5/9; given that, 4 green remain among 8 balls, so the second is green with probability 4/8. Multiplying gives (5/9)(4/8) = 20/72 = 5/18.

Question 50Counting and Probability

How many different 4-letter codes can be formed from the letters A, B, C, D, E, F if no letter may be repeated?

  1. 360 Correct answer.
  2. 15
  3. 1296
  4. 24

Explanation

This is a permutation of 4 items chosen from 6: P(6,4) = 6·5·4·3 = 360.