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← Precalculus
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Practice deliberately

Work through Precalculus unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Precalculus coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Precalculus syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use precalculus practice questions with answers.

01

Function analysis

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a function analysis problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Polynomial and rational models

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a polynomial and rational models problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Exponential and logarithmic functions

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a exponential and logarithmic functions problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Trigonometric functions

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a trigonometric functions problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Conic sections

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a conic sections problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Sequences

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a sequences problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Limits preview

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a limits preview problem from its wording, carry out the governing method, and check that the result is reasonable.

Precalculus questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: If f(x) = 2x - 5 and g(x) = x^2 + 1, what is f(g(3))?

Precalculus · Function analysis

  1. 15
  2. 5
  3. 2
  4. 26
Show the worked answer
  1. Work from the inside out. First find g(3).
  2. g(3) = 3^2 + 1 = 9 + 1 = 10.
  3. Now put 10 into f: f(10) = 2(10) - 5.
  4. 20 - 5 = 15.

Answer: 15

Question 2: Find all real zeros of p(x) = x^3 - 2x^2 - 5x + 6.

Precalculus · Polynomial and rational models

  1. -2, 1, and 3
  2. -1, 2, and 3
  3. -3, 1, and 2
  4. 1, 2, and 6
Show the worked answer
  1. Try small whole numbers that divide 6. At x = 1: 1 - 2 - 5 + 6 = 0, so x = 1 is a zero.
  2. Divide out the factor (x - 1) to get x^2 - x - 6.
  3. Factor that: x^2 - x - 6 = (x - 3)(x + 2).
  4. The zeros are x = 1, x = 3, and x = -2.

Answer: -2, 1, and 3

Question 3: What is the horizontal asymptote of f(x) = (3x^2 + 2x) / (6x^2 - 5)?

Precalculus · Polynomial and rational models

  1. y = 1/2
  2. y = 0
  3. y = 3
  4. there is none
Show the worked answer
  1. The top and the bottom have the same highest power, x^2.
  2. When the degrees match, the asymptote is the ratio of the leading coefficients.
  3. The leading coefficients are 3 on top and 6 on the bottom.
  4. 3 / 6 = 1/2, so the horizontal asymptote is y = 1/2.

Answer: y = 1/2

Question 4: Solve 5^x = 200. Give the answer to three decimal places.

Precalculus · Exponential and logarithmic functions

  1. x = 3.292
  2. x = 2.301
  3. x = 40.000
  4. x = 1.609
Show the worked answer
  1. Take the natural log of both sides: x ln 5 = ln 200.
  2. So x = ln 200 / ln 5.
  3. ln 200 is about 5.2983 and ln 5 is about 1.6094.
  4. 5.2983 / 1.6094 = 3.292 to three decimal places.

Answer: x = 3.292

Question 5: What is the exact value of cos(7 pi / 6)?

Precalculus · Trigonometric functions

  1. negative square root of 3, over 2
  2. square root of 3, over 2
  3. -1/2
  4. 1/2
Show the worked answer
  1. 7 pi / 6 is pi / 6 past pi, so the angle lands in the third quadrant.
  2. In the third quadrant cosine is negative.
  3. The reference angle is pi / 6, and cos(pi / 6) = square root of 3, over 2.
  4. Apply the negative sign: cos(7 pi / 6) = negative square root of 3, over 2.

Answer: negative square root of 3, over 2

Question 6: Where are the foci of the ellipse x^2/25 + y^2/9 = 1?

Precalculus · Conic sections

  1. (4, 0) and (-4, 0)
  2. (0, 4) and (0, -4)
  3. (5, 0) and (-5, 0)
  4. (3, 0) and (-3, 0)
Show the worked answer
  1. The larger denominator is under x^2, so the ellipse is wider than it is tall and the foci sit on the x-axis.
  2. Here a^2 = 25 and b^2 = 9.
  3. For an ellipse, c^2 = a^2 - b^2 = 25 - 9 = 16.
  4. c = 4, so the foci are (4, 0) and (-4, 0).

Answer: (4, 0) and (-4, 0)

Question 7: A geometric sequence starts at 3 and doubles each term. What is the eighth term?

Precalculus · Sequences

  1. 384
  2. 768
  3. 256
  4. 192
Show the worked answer
  1. The nth term of a geometric sequence is first term times ratio to the power (n - 1).
  2. Here the ratio is 2 and n = 8, so the power is 7.
  3. 2^7 = 128.
  4. 3 x 128 = 384.

Answer: 384

Question 8: Find the limit of (x^2 - 9) / (x - 3) as x approaches 3.

Precalculus · Limits preview

  1. 6
  2. 0
  3. 3
  4. the limit does not exist
Show the worked answer
  1. Putting x = 3 straight in gives 0/0, which tells you nothing yet.
  2. Factor the top: x^2 - 9 = (x - 3)(x + 3).
  3. Cancel the common factor (x - 3), leaving x + 3 for every x other than 3.
  4. As x gets close to 3, x + 3 gets close to 6, so the limit is 6.

Answer: 6

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Precalculus questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Function analysisIf f(x) = 2x - 5 and g(x) = x^2 + 1, what is f(g(3))?15
2Polynomial and rational modelsFind all real zeros of p(x) = x^3 - 2x^2 - 5x + 6.-2, 1, and 3
3Polynomial and rational modelsWhat is the horizontal asymptote of f(x) = (3x^2 + 2x) / (6x^2 - 5)?y = 1/2
4Exponential and logarithmic functionsSolve 5^x = 200. Give the answer to three decimal places.x = 3.292
5Trigonometric functionsWhat is the exact value of cos(7 pi / 6)?negative square root of 3, over 2
6Conic sectionsWhere are the foci of the ellipse x^2/25 + y^2/9 = 1?(4, 0) and (-4, 0)
7SequencesA geometric sequence starts at 3 and doubles each term. What is the eighth term?384
8Limits previewFind the limit of (x^2 - 9) / (x - 3) as x approaches 3.6

How the generated Precalculus sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Precalculus practice page

What does this page cover?

It covers all 7 Precalculus units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.