Pre-Calculus
Pre-calculus builds the advanced tools you need for calculus and higher math: functions, exponentials, logarithms, and sequences.
Functions
A function is a rule that assigns exactly one output to each input. We write f(x) to mean "the function f evaluated at x."
Example
f(x) = 2x + 3
f(4) = 2(4) + 3 = 8 + 3 = 11
f(−1) = 2(−1) + 3 = −2 + 3 = 1
The set of valid inputs is the domain. The set of outputs is the range. A function passes the vertical line test: any vertical line crosses its graph at most once.
Exponential and Logarithmic Functions
Exponential functions grow (or decay) by a constant factor: f(x) = a · bˣ.
A logarithm is the inverse of an exponent: log_b(x) = y means bʸ = x.
Key Rules
log(AB) = log A + log B
log(A/B) = log A − log B
log(Aⁿ) = n · log A
ln(e) = 1 (natural log)
Example
log₂(8) = 3 because 2³ = 8
Sequences & Series
An arithmetic sequence adds a constant difference each term: 3, 7, 11, 15 … (d = 4).
A geometric sequence multiplies by a constant ratio each term: 2, 6, 18, 54 … (r = 3).
| nth Term | Sum of n Terms | |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n−1)d | Sₙ = n/2 · (a₁ + aₙ) |
| Geometric | aₙ = a₁ · rⁿ⁻¹ | Sₙ = a₁(1−rⁿ)/(1−r) |
Introduction to Limits
A limit describes the value a function approaches as the input approaches some value.
Example
lim (x→2) [x² − 4]/(x − 2)
Factor: (x+2)(x−2)/(x−2) = x + 2
As x → 2: 2 + 2 = 4
Limits are the foundation of calculus. They let us work with functions at points where direct substitution is undefined.
Quick Quiz
Test what you just learned. Choose the best answer for each question.