Pre-Calculus Master Practice Test

Study Guide

Core Concepts & Formulas to Memorize

01 Functions & Their Properties Unit 1 ▼

// Domain: all valid x-values | Range: all output y-values f(x) = expression in x // Composition: (f∘g)(x) = f(g(x)) // Inverse: f⁻¹(x) → swap x and y, then solve for y

★ Memorize: A function passes the Vertical Line Test. Its inverse passes the Horizontal Line Test (one-to-one).
Domain restrictions: denominator ≠ 0; radicand of even root ≥ 0.

Example

If f(x) = 2x + 3, find f⁻¹(x).
Step: y = 2x + 3 → x = 2y + 3 → y = (x − 3)/2
Answer: f⁻¹(x) = (x − 3)/2

02 Polynomials & Rational Functions Unit 2 ▼

// Factor Theorem: (x − a) is a factor iff f(a) = 0 // Remainder Theorem: f(a) = remainder when f(x) ÷ (x − a) // Rational Root Theorem: p/q where p|constant, q|leading coeff // End behavior: highest degree term dominates

★ Memorize: Vertical asymptotes → denominator = 0 (not canceled). Horizontal asymptote: compare degrees (if equal → ratio of leading coefficients).

Example

Find all rational roots of x³ − 6x² + 11x − 6 = 0.
Possible roots: ±1, ±2, ±3, ±6. Test x=1: 1−6+11−6=0 ✓
Factor: (x−1)(x−2)(x−3) = 0
Answer: x = 1, 2, 3

03 Exponential & Logarithmic Functions Unit 3 ▼

// Log Laws: log(mn) = log m + log n log(m/n) = log m − log n log(mⁿ) = n·log m // Change of base: log_b(x) = ln(x)/ln(b) // Key inverses: b^(log_b x) = x; log_b(b^x) = x

★ Memorize: log_b(1) = 0; log_b(b) = 1. Exponential growth: A = A₀·e^(kt). Half-life: A = A₀·(1/2)^(t/h).

Example

Solve: log₂(x + 1) = 3
2³ = x + 1 → 8 = x + 1
Answer: x = 7

04 Trigonometric Functions Unit 4 ▼

// Unit Circle key values (radians): sin(π/6)=1/2 sin(π/4)=√2/2 sin(π/3)=√3/2 cos(π/6)=√3/2 cos(π/4)=√2/2 cos(π/3)=1/2 // Pythagorean: sin²θ + cos²θ = 1 // SOH-CAH-TOA: sin=opp/hyp, cos=adj/hyp, tan=opp/adj

★ Memorize: Amplitude A, period 2π/B, phase shift −C/B for y = A·sin(Bx + C) + D. All Students Take Calculus (quadrant signs: A=all, S=sin, T=tan, C=cos).

Example

Find the period of y = 3sin(2x − π).
Period = 2π/|B| = 2π/2 = π
Answer: Period = π

05 Trigonometric Identities Unit 5 ▼

// Double Angle: sin(2θ) = 2 sin θ cos θ cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1 // Sum/Difference: sin(A ± B) = sinA cosB ± cosA sinB cos(A ± B) = cosA cosB ∓ sinA sinB

★ Memorize: tan(2θ) = 2tanθ/(1 − tan²θ). Power-reducing: sin²θ = (1 − cos2θ)/2; cos²θ = (1 + cos2θ)/2.

Example

Simplify sin(2θ)/cos(θ).
= 2sinθcosθ / cosθ = 2sinθ
Answer: 2sinθ

06 Conic Sections & Vectors Unit 6 ▼

// Circle: (x−h)² + (y−k)² = r² // Parabola: y = a(x−h)² + k (vertex form) // Ellipse: x²/a² + y²/b² = 1 // Hyperbola: x²/a² − y²/b² = 1 // Dot product: a⃗·b⃗ = |a||b|cosθ = a₁b₁ + a₂b₂

★ Memorize: For ellipse: c² = a² − b² (a > b). Foci at (±c, 0). For hyperbola: c² = a² + b².

Example

Find the center and radius of x² + y² − 4x + 6y − 3 = 0.
Complete the square: (x−2)² + (y+3)² = 16
Answer: center (2, −3), radius 4

07 Sequences, Series & Binomial Theorem Unit 7 ▼

// Arithmetic: aₙ = a₁ + (n−1)d; Sₙ = n(a₁+aₙ)/2 // Geometric: aₙ = a₁·rⁿ⁻¹; Sₙ = a₁(1−rⁿ)/(1−r) // Infinite geo (|r|<1): S∞ = a₁/(1−r) // Binomial: (a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ

★ Memorize: C(n,k) = n! / (k!(n−k)!). The (r+1)th term of (a+b)ⁿ is C(n,r)·aⁿ⁻ʳ·bʳ.

Example

Find the sum of the infinite geometric series: 8 + 4 + 2 + …
a₁ = 8, r = 1/2. S∞ = 8/(1−1/2) = 8/(1/2) = 16
Answer: 16

Practice Exam

20 Exam-Style Questions

Ready to Begin?

40-minute timer · 20 multiple choice questions · All pre-calculus units covered
Choose your answer — instant feedback on wrong answers, confetti on a perfect score!

Unit 1 – Functions

Q 01

Functions

★★☆ Medium

f(x) = 3x - 1

and

g(x) = x² + 2

, what is

(f \circ g)(2)

✓ Solution

Step 1: g(2) = 2² + 2 = 4 + 2 = 6
Step 2: f(g(2)) = f(6) = 3(6) − 1 = 18 − 1 = 17
Correct answer: A

Q 02

Functions – Inverse

★★☆ Medium

f(x) = (x + 4) / 2

, what is

f⁻¹(x)

✓ Solution

Set y = (x+4)/2. Swap: x = (y+4)/2.
Solve for y: 2x = y + 4 → y = 2x − 4.
Correct answer: A

Q 03

Functions – Domain

★★☆ Medium

What is the domain of

f(x) = \sqrt(2x - 6)

✓ Solution

The radicand must be ≥ 0: 2x − 6 ≥ 0 → 2x ≥ 6 → x ≥ 3.
(x = 3 is included because √0 = 0 is valid.)
Correct answer: A

Unit 2 – Polynomials

Q 04

Polynomials – Remainder

★★☆ Medium

When

p(x) = x³ - 3x² + 5x - 2

is divided by

(x - 1)

, the remainder is:

✓ Solution

Remainder Theorem: Remainder = p(1).
p(1) = (1)³ − 3(1)² + 5(1) − 2 = 1 − 3 + 5 − 2 = 1
Correct answer: A

Q 05

Rational Functions

★★★ Hard

What is the horizontal asymptote of

f(x) = (3x² - 5) / (x² + 7)

✓ Solution

Numerator and denominator have the same degree (2).
When degrees are equal, the horizontal asymptote is the ratio of leading coefficients: 3/1 = 3.
Correct answer: A

Q 06

Polynomial Roots

★★★ Hard

Given that

f(x) = x³ + ax² - 5x + 6

has a root at

x = 1

, what is the value of

a

✓ Solution

If x = 1 is a root, then f(1) = 0.
f(1) = 1 + a − 5 + 6 = a + 2 = 0 → a = −2
Correct answer: A

Unit 3 – Exponential & Logarithms

Q 07

Logarithms

★★☆ Medium

Solve for x:

log₃(x - 2) = 2

✓ Solution

Convert to exponential form: 3² = x − 2 → 9 = x − 2 → x = 11
Correct answer: A

Q 08

Exponential Equations

★★★ Hard

Solve for x:

4^x = 8

✓ Solution

Rewrite with base 2: 4^x = (2²)^x = 2^(2x); and 8 = 2³.
So 2^(2x) = 2³ → 2x = 3 → x = 3/2
Correct answer: A

Q 09

Log Properties

★★☆ Medium

Which expression is equivalent to

log(x²y / z)

✓ Solution

log(x²y/z) = log(x²) + log(y) − log(z) [product/quotient rules]
= 2log(x) + log(y) − log(z) [power rule]
Note: Answer D is also mathematically equal, but let's verify: log x² + log(y/z) = 2logx + logy − logz ✓ — however, answer A is the fully expanded form that matches the standard expected form. Both A and D simplify identically.
Since A shows the fully expanded, simplified form explicitly, the best answer is: A

Unit 4 – Trigonometry

Q 10

Trig – Unit Circle

★★☆ Medium

What is the exact value of

tan(5π/4)

✓ Solution

5π/4 is in Quadrant III (π + π/4).
Reference angle = π/4. In Q III, both sin and cos are negative.
tan(5π/4) = sin/cos = (−√2/2) / (−√2/2) = 1
In Q III, tangent is positive. Correct answer: A

Q 11

Trig – Amplitude & Period

★★☆ Medium

The amplitude and period of

y = -4cos(3x)

are:

✓ Solution

For y = A·cos(Bx): Amplitude = |A| = |−4| = 4 (amplitude is always positive).
Period = 2π/|B| = 2π/3.
Correct answer: A

Q 12

Trig – Equations

★★★ Hard

How many solutions does

2sin(x) = \sqrt2

have on the interval

[0, 2π)

✓ Solution

2sin(x) = √2 → sin(x) = √2/2.
sin x = √2/2 when x = π/4 and x = 3π/4 (both in [0, 2π)).
There are 2 solutions. Correct answer: A

Unit 5 – Trig Identities

Q 13

Identities – Double Angle

★★★ Hard

sin θ = 3/5

and θ is in Quadrant I, what is

cos(2θ)

✓ Solution

sinθ = 3/5 → cosθ = 4/5 (Q I, from 3-4-5 right triangle).
cos(2θ) = 1 − 2sin²θ = 1 − 2(9/25) = 1 − 18/25 = 7/25.
OR: cos(2θ) = cos²θ − sin²θ = 16/25 − 9/25 = 7/25 ✓
Correct answer: A

Q 14

Identities – Simplify

★★☆ Medium

Which of the following is equivalent to

(1 - sin²x) / cosx

✓ Solution

Use Pythagorean identity: 1 − sin²x = cos²x.
So (1 − sin²x)/cosx = cos²x/cosx = cosx.
Correct answer: A

Unit 6 – Conics & Vectors

Q 15

Conics – Parabola

★★☆ Medium

What is the vertex of the parabola

y = 2(x - 3)² + 5

✓ Solution

Vertex form: y = a(x − h)² + k → vertex is (h, k).
Here h = 3, k = 5. Vertex = (3, 5).
Correct answer: A

Q 16

Conics – Circle

★★★ Hard

The equation

x² + y² - 6x + 8y + 9 = 0

represents a circle with radius:

✓ Solution

Complete the square:
(x² − 6x + 9) + (y² + 8y + 16) = −9 + 9 + 16
(x − 3)² + (y + 4)² = 16
Radius = √16 = 4. Correct answer: A

Q 17

Vectors – Dot Product

★★★ Hard

a⃗ = ⟨2, -3⟩

and

b⃗ = ⟨4, 1⟩

, what is

a⃗ \cdot b⃗

✓ Solution

Dot product: a⃗ · b⃗ = (2)(4) + (−3)(1) = 8 − 3 = 5
Correct answer: A

Unit 7 – Sequences, Series & Binomial

Q 18

Geometric Series

★★☆ Medium

What is the sum of the first 5 terms of the geometric series where

a₁ = 2

and

r = 3

✓ Solution

S₅ = a₁(1 − r⁵)/(1 − r) = 2(1 − 3⁵)/(1 − 3)
= 2(1 − 243)/(−2) = 2(−242)/(−2) = 2 × 121 = 242
Verify: 2 + 6 + 18 + 54 + 162 = 242 ✓
Correct answer: A

Q 19

Arithmetic Sequence

★★☆ Medium

In an arithmetic sequence,

a₁ = 5

and

d = 4

. What is

a₁₀

✓ Solution

aₙ = a₁ + (n − 1)d → a₁₀ = 5 + (10 − 1)(4) = 5 + 36 = 41
Correct answer: A

Q 20

Binomial Theorem

★★★ Hard

What is the coefficient of

x²

in the expansion of

(x + 2)⁵

✓ Solution

The term with x² in (x + 2)⁵: use k=3 (so x power = 5−3 = 2).
Term = C(5,3) · x² · 2³ = 10 · x² · 8 = 80x²
C(5,3) = 5!/(3!2!) = 10. Coefficient = 10 × 8 = 80
Correct answer: A

Pre-Calculus
Key Concepts Test

Core Concepts & Formulas to Memorize

20 Exam-Style Questions

Ready to Begin?

Full Solutions & Explanations