Pre-calculus cheat sheet

Functions and domains

A function assigns one output to each permitted input. The domain must respect restrictions in its formula or context.

For real outputs, √(x − 2) requires x ≥ 2.

Transformations

Changes inside a function’s input act differently from changes to its output. Track the sign of a horizontal shift carefully.

y = f(x − h) shifts the graph h units right when h > 0.

Logarithms

A logarithm is an exponent: log_b(a) = c means bᶜ = a, with b > 0, b ≠ 1 and a > 0 in the real setting.

log₁₀(1000) = 3.

Unit circle

A point at angle θ on the unit circle has coordinates (cos θ, sin θ). The angle convention must be clear.

At θ = π/2, the point is (0, 1).

Function composition

The composition f∘g means apply g first, then f. Its domain includes only inputs for which g is defined and its output is accepted by f.

If f(x) = 2x and g(x) = x + 3, then f(g(x)) = 2x + 6.

Inverse functions

An inverse reverses a one-to-one function on the chosen domain. Interchanging input and output is a method to derive the rule, but domain restrictions still matter.

For f(x) = 2x + 3 on the real numbers, f⁻¹(x) = (x − 3)/2.

Arithmetic and geometric sequences

Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. The first index must be stated before using a general term.

With first term a₁, arithmetic aₙ = a₁ + (n − 1)d; geometric aₙ = a₁rⁿ⁻¹.

Trigonometric identities

The identity sin²θ + cos²θ = 1 follows from the unit circle. An identity holds throughout its permitted domain, unlike an equation solved only at selected values.

tan θ = sin θ/cos θ requires cos θ ≠ 0.

Worked example: exponential growth

A model starts at 200 and grows by 5% per step. After three steps its value is 200(1.05)³ = 231.525, or about 231.5 at the stated precision.

Repeated percentage growth multiplies each step; it does not add 5 units or simply add 15%.

Common mistake: logarithms distribute over addition

Logarithm laws convert products into sums, not sums into separate logarithms. Check positive real arguments and a valid base.

log(ab) = log a + log b for positive a and b, but log(a + b) generally differs from log a + log b.

Check your understanding

If f(x) = x + 1 and g(x) = x², what is g(f(2))?

9: f(2) = 3, then g(3) = 9.

Curriculum connections

Sources