Functions & Graphs

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).

f: Domain → Range
f(x) = expression involving x

Example: f(x) = 2x + 3

f(0) = 3, f(1) = 5, f(−2) = −1

This is a linear function — its graph is a straight line with slope 2 and y-intercept 3.

Domain and Range

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).

Piecewise Functions

Defined by different rules on different intervals. Understanding piecewise functions prepares you for limits and continuity in calculus.

Transformations

Given a function f(x), you can shift, stretch, compress, and reflect its graph systematically:

Vertical shift: f(x) + k (up k units) or f(x) − k (down)
Horizontal shift: f(x − h) (right h units) or f(x + h) (left)
Vertical stretch/compress: a·f(x) (stretch if |a| > 1, compress if |a| < 1)
Reflection: −f(x) (over x-axis) or f(−x) (over y-axis)

These transformations generalize to all functions — from trig functions to probability distributions.

Composition & Inverses

Composition

(f ∘ g)(x) = f(g(x))

Apply g first, then f. Composition is the basis of the chain rule in calculus.

Example: f(x) = x², g(x) = 3x + 1

(f ∘ g)(x) = f(3x + 1) = (3x + 1)²

(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).

f(x) = eˣ → f'(x) = eˣ (the only function equal to its own derivative!)

Logarithmic Functions: f(x) = log_a(x)

The logarithm is the inverse of the exponential: log_a(x) = y means aʸ = x.

Key Properties:
log(ab) = log(a) + log(b)
log(a/b) = log(a) − log(b)
log(aⁿ) = n·log(a)
log_a(1) = 0, log_a(a) = 1

Logarithmic scales appear in statistics (log transformations), the Richter scale for earthquakes, and the decibel scale for sound. The natural logarithm ln(x) is essential in integration and differential equations.

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.

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