TL;DR: Pre-calculus bridges algebra and trig into the foundation you need for calculus. The key topics are: functions (domain, range, transformations), polynomial and rational functions, exponential and logarithmic functions, trigonometry review, sequences and series, and an intro to limits. If you're solid on these, calculus won't feel like hitting a wall.

Why Pre-Calculus Matters More Than You Think

A lot of students treat pre-calculus like filler — something to get through before the "real" math starts. That's a mistake.

Pre-calc is where you build every single skill calculus assumes you already have. When students struggle in Calc I, it's almost never because they can't understand derivatives. It's because they're shaky on algebra, functions, or trig from pre-calc.

Think of pre-calc as boot camp. The fitter you are going in, the less painful calculus will be.

This guide covers everything you should know before walking into your first calculus class. If you're reviewing for a pre-calc final, prepping for calc, or just want to fill gaps, this is for you.

1. Functions: The Foundation of Everything

Functions are the single most important pre-calc concept. Calculus is entirely about functions — taking their derivatives, integrating them, analyzing their behavior.

What's a Function?

A function is a rule that takes an input (x) and gives exactly one output (y). Written as f(x) = something.

Example: f(x) = 2x + 3

Input x = 4 → Output f(4) = 2(4) + 3 = 11
Each input gives one and only one output

Domain and Range

Domain: All possible input values (x-values) Range: All possible output values (y-values)

Most common domain restrictions:

Can't divide by zero (denominators ≠ 0)
Can't take square root of negatives (expression under √ ≥ 0)
Can't take log of zero or negatives (argument > 0)

Example: f(x) = 1/(x - 3)

Domain: all real numbers except x = 3
Because plugging in 3 gives 1/0 (undefined)

Function Types You Need to Know

Function Type	General Form	Shape
Linear	f(x) = mx + b	Straight line
Quadratic	f(x) = ax² + bx + c	Parabola
Polynomial	f(x) = aₙxⁿ + ... + a₁x + a₀	Curves
Rational	f(x) = p(x)/q(x)	Curves with asymptotes
Exponential	f(x) = aˣ	Growth/decay curve
Logarithmic	f(x) = log(x)	Inverse of exponential
Trigonometric	f(x) = sin(x), cos(x), etc.	Waves

Function Operations

You should be able to:

Add/subtract/multiply/divide functions: (f + g)(x) = f(x) + g(x)
Compose functions: (f ∘ g)(x) = f(g(x)) — plug g into f
Find inverses: Switch x and y, solve for y

Composition example: If f(x) = x² and g(x) = x + 1:

(f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)² = x² + 2x + 1
(g ∘ f)(x) = g(f(x)) = g(x²) = x² + 1

Note: f ∘ g ≠ g ∘ f in most cases!

Transformations

When you see f(x) = a · f(b(x - h)) + k, here's what each part does:

Parameter	Effect
a	Vertical stretch (
b	Horizontal stretch (
h	Shift right (h > 0) or left (h < 0)
k	Shift up (k > 0) or down (k < 0)

Remember: Horizontal transformations do the opposite of what you'd expect. (x - 3) shifts RIGHT 3, not left.

2. Polynomial Functions

Key Concepts

Degree: The highest power of x. Tells you:

Maximum number of zeros (x-intercepts)
Maximum number of turning points (degree - 1)
End behavior

End behavior:

Odd degree: starts low, ends high (positive leading coefficient) or starts high, ends low (negative)
Even degree: both ends up (positive) or both ends down (negative)

Factoring (You NEED This)

Calculus assumes you can factor quickly. Practice these:

Common factor: 3x² + 6x = 3x(x + 2)
Difference of squares: x² - 9 = (x + 3)(x - 3)
Trinomial: x² + 5x + 6 = (x + 2)(x + 3)
Grouping: For 4-term polynomials
Sum/difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)

The Rational Root Theorem

If a polynomial has rational zeros, they'll be of the form ±(factor of constant term)/(factor of leading coefficient). This helps you test possible roots.

3. Rational Functions

Rational functions are fractions where both numerator and denominator are polynomials: f(x) = p(x)/q(x).

Asymptotes

Vertical asymptotes: Where the denominator = 0 (and numerator ≠ 0). The function shoots to infinity.

Horizontal asymptotes: What happens as x → ∞:

If degree of numerator < degree of denominator → y = 0
If degrees are equal → y = ratio of leading coefficients
If degree of numerator > degree of denominator → no horizontal asymptote (may have slant/oblique asymptote)

Example: f(x) = (2x + 1)/(x - 3)

Vertical asymptote: x = 3
Horizontal asymptote: y = 2/1 = 2 (degrees equal, leading coefficients 2 and 1)

4. Exponential and Logarithmic Functions

Exponential Functions: f(x) = aˣ

When a > 1: exponential growth
When 0 < a < 1: exponential decay
The most important base: e ≈ 2.718 (the natural exponential)

Logarithmic Functions: f(x) = logₐ(x)

Logs are the inverse of exponentials. If aʸ = x, then logₐ(x) = y.

Common logs:

log(x) = log₁₀(x) — base 10
ln(x) = logₑ(x) — natural log (base e)

Log Rules (Memorize These)

Product rule: log(ab) = log(a) + log(b)
Quotient rule: log(a/b) = log(a) - log(b)
Power rule: log(aⁿ) = n·log(a)
Change of base: logₐ(b) = log(b)/log(a)

Solving Exponential and Log Equations

Exponential equation: 3ˣ = 81

Take log of both sides: x·log(3) = log(81)
x = log(81)/log(3) = 4

Log equation: log₂(x + 3) = 5

Rewrite as exponential: 2⁵ = x + 3
32 = x + 3
x = 29

5. Trigonometry Review

If you haven't already, check out our complete trigonometry guide. Here's the quick review:

Must-Know Trig Facts

SOH CAH TOA for right triangles
Unit circle values for 0°, 30°, 45°, 60°, 90°
sin²θ + cos²θ = 1
Graphs of sin and cos (period, amplitude, phase shift)
Radian/degree conversion
Law of Sines and Law of Cosines

Trig Identities for Calculus

These specific identities come up in Calc I and II:

Pythagorean: sin²θ + cos²θ = 1
Double angle: sin(2θ) = 2sinθcosθ
Half angle: sin²θ = (1 - cos2θ)/2
Power reducing: cos²θ = (1 + cos2θ)/2

The half-angle and power-reducing identities become critical in Calc II integration. Start getting familiar now.

6. Sequences and Series

Arithmetic Sequences

Common difference (d) between terms
aₙ = a₁ + (n-1)d
Sum: Sₙ = n(a₁ + aₙ)/2

Geometric Sequences

Common ratio (r) between terms
aₙ = a₁ · rⁿ⁻¹
Sum: Sₙ = a₁(1 - rⁿ)/(1 - r) when r ≠ 1
Infinite sum (|r| < 1): S = a₁/(1 - r)

Why This Matters for Calculus

Series are the foundation of Taylor series, power series, and convergence tests in Calc II. A solid understanding of geometric series is particularly important.

7. Intro to Limits (The Bridge to Calculus)

Some pre-calc courses introduce limits. Even if yours didn't, understanding the concept gives you a head start.

What's a Limit?

A limit asks: "What value does f(x) approach as x gets closer and closer to some number?"

Written as: lim(x→a) f(x) = L

Example: lim(x→2) (x² + 1) = 2² + 1 = 5

For continuous functions, you can often just plug in the value. Limits get interesting when plugging in doesn't work (like 0/0 situations).

When Direct Substitution Fails

If you get 0/0, try:

Factor and cancel: lim(x→3) (x² - 9)/(x - 3) = lim(x→3) (x+3)(x-3)/(x-3) = lim(x→3) (x+3) = 6
Rationalize: Multiply by conjugate for expressions with square roots
L'Hôpital's rule: (You'll learn this in calculus — take derivative of top and bottom)

Limits at Infinity

What happens as x gets really big?

lim(x→∞) 1/x = 0 (gets smaller and smaller)
lim(x→∞) x² = ∞ (grows without bound)

8. Other Topics to Review

Coordinate Geometry

Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²)
Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂-y₁)/(x₂-x₁)
Equations of lines: y = mx + b, y - y₁ = m(x - x₁)

Systems of Equations

Substitution method
Elimination method
Matrix methods (for 3+ variables)

Inequalities and Absolute Value

|x| < a means -a < x < a
|x| > a means x < -a OR x > a
Polynomial inequalities: use sign charts

Conic Sections

Circle: (x-h)² + (y-k)² = r²
Ellipse, parabola, hyperbola (know the general forms)

Pre-Calculus Self-Assessment Checklist

Before starting calculus, make sure you can:

If you checked fewer than 7, spend extra time reviewing those weak areas. Gradily can help you practice specific problem types and get step-by-step explanations for anything that's not clicking.

Study Plan: Pre-Calc Review in One Week

If you're cramming before calculus starts:

Day	Topic	Time
1	Functions: domain, range, operations	2 hours
2	Polynomials: factoring, zeros, end behavior	2 hours
3	Rational functions + asymptotes	1.5 hours
4	Exponential + logarithmic functions	2 hours
5	Trigonometry review	2 hours
6	Sequences, series, limits intro	1.5 hours
7	Mixed practice problems	2 hours

Focus on understanding, not memorizing. If you can explain why something works, you'll remember it when it matters.

Wrapping Up

Pre-calculus is the foundation that makes calculus manageable. Every minute you spend solidifying these concepts now saves you hours of frustration later.

The students who do well in calculus aren't necessarily the smartest — they're the ones who walked in with strong pre-calc fundamentals. Functions, factoring, trig, and logs are your tools. Sharpen them now.

Need help reviewing specific pre-calc topics? Gradily can walk you through problems step by step, from basic function analysis to tricky trig identities.

Ready for what comes next? Check out our calculus for beginners guide. Or brush up on trigonometry and algebra first.

Editorial Standards