Functions are the universal language of mathematics — learn to speak it fluently.
What Is a Function?
A function is a rule that assigns to each input exactly one output. We write f(x) for the output of function f at input x. The key property: every input has exactly one output (the vertical line test).
The domain is the set of all valid inputs. The range is the set of all possible outputs.
Polynomial functions: Domain = all real numbers
Rational functions: Exclude values where the denominator = 0 (see rational expressions)
Square root functions: Require the radicand ≥ 0
Logarithmic functions: Require the argument > 0
Example: f(x) = √(x − 2)
Domain: x − 2 ≥ 0 → x ≥ 2, i.e., [2, ∞)
Range: [0, ∞) since √ always gives non-negative results
Types of Functions
Linear: f(x) = mx + b
Graph is a straight line. Slope m = rise/run = rate of change. The concept of slope becomes the derivative in calculus.
Quadratic: f(x) = ax² + bx + c
Graph is a parabola. Vertex at (−b/2a, f(−b/2a)). Opens up if a > 0, down if a < 0. Zeros found by solving quadratic equations. Parabolas are a type of conic section.
Polynomial: f(x) = aₙxⁿ + ⋯ + a₁x + a₀
Smooth, continuous curves. Degree n means at most n real zeros and n − 1 turning points. See Polynomials & Factoring for detailed analysis.
Rational: f(x) = P(x)/Q(x)
May have vertical asymptotes (where Q = 0), horizontal asymptotes (end behavior), and holes (where P and Q share a factor).
(g ∘ f)(x) = g(x²) = 3x² + 1 — note: f ∘ g ≠ g ∘ f in general!
Inverse Functions
The inverse f⁻¹ "undoes" f: if f(a) = b, then f⁻¹(b) = a. To find f⁻¹, swap x and y, then solve for y.
f(f⁻¹(x)) = f⁻¹(f(x)) = x
Example: Find f⁻¹ for f(x) = (2x − 3)/5
y = (2x − 3)/5 → swap: x = (2y − 3)/5 → 5x = 2y − 3 → y = (5x + 3)/2
f⁻¹(x) = (5x + 3)/2
A function has an inverse only if it's one-to-one (passes the horizontal line test). The inverse trig functions require restricting the domain to achieve this.
Exponential & Logarithmic Functions
Exponential Functions: f(x) = aˣ
These model explosive growth (a > 1) or decay (0 < a < 1). The most important base is e ≈ 2.71828 (Euler's number).
Functions are the central object of study in all of higher mathematics. Calculus studies how functions change. Linear algebra studies functions between vector spaces. Differential equations describe functions through their derivatives.