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Praxis Mathematics: Functions and Calculus

Functions, their graphs, and introductory rates and accumulation for secondary mathematics, including decisions about student work.

Teacher certification exams are administered by testing organizations. FreeExam is not affiliated with those organizations, and these questions are original practice, not an official test.

Praxis is a trademark of Educational Testing Service. FreeExam is not affiliated with Educational Testing Service, and these questions are original practice, not an official test.

  1. 1.If f(x) = 2x + 1, what is f(3)?

  2. 2.Which set is the domain of f(x) = √(x − 1)?

  3. 3.What is the range of f(x) = x^2 for real x?

  4. 4.A student includes 0 in the domain of f(x) = 1/x. Which question targets the error?

  5. 5.f(x) = x + 2 and g(x) = 3x. What is f(g(2))?

  6. 6.Which function is the inverse of f(x) = 3x?

  7. 7.A population doubles every year and starts at 5. Which formula matches that growth after t years?

  8. 8.A student shifts y = x^2 to the right by 2 and writes y = (x + 2)^2. What is the correction?

  9. 9.The graph of y = 3x^2 is which change from y = x^2?

  10. 10.On the interval from x = 1 to x = 4, f(1) = 2 and f(4) = 8. What is the average rate of change?

  11. 11.What is log base 10 of 1,000?

  12. 12.A student says log(100) + log(10) = log(110), adding the inputs. Which identity should the teacher set beside that claim?

  13. 13.Which graph feature belongs to y = −x^2 + 4?

  14. 14.f(x) = 2^x. Which value is f(0)?

  15. 15.As x approaches 2, what is the limit of (x^2 − 4)/(x − 2), if the limit is taken and x is not set equal to 2?

  16. 16.A student says the derivative of x^2 at x = 3 is 9, because they squared 3. Which meaning of the derivative corrects that?

  17. 17.A car's position in meters is s(t) = 4t. What is its velocity?

  18. 18.The area under y = 3 from x = 0 to x = 4 is which value?

  19. 19.Which function grows faster in the long run, y = 100x or y = 2^x?

  20. 20.A student reflects y = 2^x across the y-axis and writes y = 2^(−x), then says the new function still grows as x increases. What is the accurate description?

  21. 21.If f(x) = x − 4, which input gives output 0?

  22. 22.g(x) = |x − 2|. What is g(0)?

  23. 23.Which function is linear?

  24. 24.A class confuses f(x + 1) with f(x) + 1 for f(x) = x^2. Which pair of values separates them?

  25. 25.The point (2, 5) lies on the graph of a function f. Which statement must be true?

  26. 26.Which function is one-to-one on all real numbers?

  27. 27.What is the y-intercept of f(x) = −2x + 6?

  28. 28.A student finds the inverse of y = x + 5 by changing the sign, writing y = x − 5, and stops without swapping. The formula happens to be the inverse. What check still belongs in the lesson?

  29. 29.h(x) = (x + 1)/(x − 3). Which input is excluded?

  30. 30.A bacteria count is 200 × (1/2)^(t/3), with t in hours. What does the 3 in the exponent describe?

  31. 31.What is the slope of the secant line of f(x) = x^2 from x = 1 to x = 3?

  32. 32.A student estimates the derivative of a curve at a point by using a secant through that point and a second point far away. What improvement fits the definition?

  33. 33.Water flows into a tank at 2 liters per minute. How much is added from t = 0 to t = 5?

  34. 34.Which expression is log base 2 of 8?

  35. 35.f(x) = x^2 − 4x + 1. What is the axis of symmetry?

  36. 36.A student says a function cannot have the same output twice. Which example should the teacher offer?

  37. 37.Which formula is an exponential decay?

  38. 38.f(x) = 5 − x. What is f(f(1))?

  39. 39.The graph of y = |x| is shifted up 3 units. Which equation matches?

  40. 40.A student reads the difference quotient [f(3 + h) − f(3)]/h as the slope of a secant, then substitutes h = 0 immediately and gets 0/0. What should happen before substituting 0?

  41. 41.If the derivative of a function is negative on an interval, what is the function doing there?

  42. 42.f'(x) = 0 at x = 2, and f' changes from positive to negative there. What does the graph do at x = 2?

  43. 43.A rectangle under a constant rate has height equal to the rate and width equal to the time. Which calculus idea is that picture?

  44. 44.A student computes the average rate of change of f(x) = x^2 from 0 to 2 as f'(2) = 4. What is the distinction?

  45. 45.Which point is on y = log base 10 of x?

  46. 46.f(x) = 1/(x − 1) and g(x) = x + 1. For which x is f(g(x)) undefined?

  47. 47.A line has slope 0. Which description matches?

  48. 48.A student graphs y = √x and includes negative x-values because the parabola y = x^2 has them. What distinction helps?

  49. 49.What is the concavity description if f''(x) > 0 on an interval?

  50. 50.Position is s(t) = t^2, in meters, with t in seconds. What is the velocity at t = 3?

  51. 51.Which limit equals 5? The expression (x^2 − 25)/(x − 5) as x approaches 5.

  52. 52.A teacher wants a function with a jump at x = 1. Which example works?

  53. 53.Which statement about y = ln x is correct?

  54. 54.A student solves 2^x = 8 by dividing 8 by 2 and getting x = 4. What rewrite fixes the meaning?

  55. 55.f(x) = (x − 2)^2 + 5. What is the minimum value?

  56. 56.Which transformation takes y = f(x) to y = f(x) − 4?

  57. 57.If f'(x) > 0 and f''(x) < 0 on an interval, which sketch fits?

  58. 58.The integral of a velocity from t = 1 to t = 4 gives which quantity?

  59. 59.Which pair shows inverse functions?

  60. 60.A student says the range of y = 2^x is all real numbers because the graph goes up forever. What boundary is missing?

  61. 61.What is the average rate of change of f(x) = 3x − 1 from x = 2 to x = 6?

  62. 62.Which function has a vertical asymptote at x = −2?

  63. 63.f(x) = x^3 − x. What is f(−1)?

  64. 64.A student finds a critical point where f'(x) = 0 and immediately calls it a maximum. What extra information is needed?

  65. 65.Which equation has the same graph as y = log base 2 of x reflected across the line y = x?

  66. 66.From x = a to x = b, a definite integral of a rate equals which comparison of an antiderivative F?

  67. 67.A quantity is modeled by P(t) = 50(1.2)^t. What is P(0)?

  68. 68.Which input makes log base 3 of x equal 2?

  69. 69.The tangent to y = x^2 at x = 1 has slope 2. Which point-slope line is that tangent?

  70. 70.A student says a continuous function on a closed interval must be differentiable there. Which counterexample fits?

  71. 71.What is the horizontal asymptote of y = (2x + 1)/(x − 4), if there is one?

  72. 72.f(x) = −3x + 9. For which x is f(x) ≥ 0?

  73. 73.Which description matches a relative maximum of a smooth function?

  74. 74.The half-life of a substance is 5 years, and the start is 80 grams. Which expression gives the amount after t years?

  75. 75.What is the derivative of f(x) = 7 at any x?

  76. 76.A student differentiates x^3 as 3x, dropping the remaining power. What is the derivative?

  77. 77.Which function is even? An even function satisfies f(−x) = f(x).

  78. 78.If f(x) = 2x and g(x) = x − 3, what is g(f(4))?

  79. 79.A tangent line touches y = x^2 at (2, 4) and has slope 4. Which nearby feature is that slope?

  80. 80.A student integrates a speed that is always positive and gets a negative "distance." What wording fixes the interpretation?

  81. 81.What is the range of f(x) = −x^2 + 9?

  82. 82.Which equation is the inverse of y = x^3?

  83. 83.A population is 100(1.05)^t. What does 1.05 mean?

  84. 84.A student finds lim (x → 0) of sin x / x by substituting and stopping at 0/0. In a secondary introduction that states the standard limit, what value should the lesson connect to that form?

  85. 85.f(x) = 4/x. What happens to f(x) as x approaches infinity?

  86. 86.Which expression equals log base 5 of 25 minus log base 5 of 5?

  87. 87.A quadratic has zeros at x = 1 and x = 6. Which form shows those zeros?

  88. 88.A student says the derivative of a product fg is f' g'. For f(x) = x and g(x) = x, what comparison shows the error?

  89. 89.What is the domain of f(x) = √(4 − x)?

  90. 90.An object's acceleration is the derivative of its velocity. If velocity is v(t) = 6t, what is the acceleration?

  91. 91.Which function decreases as x increases through all real numbers?

  92. 92.A teacher asks why the derivative of sin x is not simply sin x. Which limiting idea is the honest answer at this level?

  93. 93.f(x) = 2(x − 1)^2 + 3. What is f(1)?

  94. 94.Which limit is infinite? Consider 1/(x − 2)^2 as x approaches 2.

  95. 95.A linear function satisfies f(0) = 2 and f(3) = 8. What is f(x)?

  96. 96.A student uses the power rule on e^x and gets x e^(x − 1). What is the derivative of e^x?

  97. 97.Which composition equals (2x + 1)^2 if the outer function is squaring?

  98. 98.On [1, 5], a continuous function has f(1) = −2 and f(5) = 4. What does the intermediate value theorem guarantee?

  99. 99.What is the slope of y = 5 − 4x?

  100. 100.A student says a horizontal asymptote cannot be crossed. Which correction is fair?

  101. 101.The area of a square is A(s) = s^2. What is A'(s), and what does it mean?

  102. 102.Which function has domain all real numbers and range y ≥ 2?

  103. 103.A car slows so that its velocity changes from 20 meters per second to 8 meters per second in 4 seconds. What is the average acceleration?

  104. 104.A student cancels x in (x^2 + x)/x and gets x + 1, then uses the result at x = 0. What should the teacher say?

  105. 105.Which point is the vertex of y = (x + 4)^2 − 1?

  106. 106.If f'(x) = 2 for every x, which function could f be?

  107. 107.The graph of a function crosses the x-axis at x = −3. Which statement is true?

  108. 108.A student estimates the area under a positive curve with one rectangle that is too tall for the whole interval. Which refinement matches the integral idea?