Functions And Graphs

Level 1 - Calculus topic page in Foundations And Limits.

Principle

A function assigns each allowed input exactly one output. Graphs turn that assignment into geometry: the point \((x,f(x))\) records the input and output together. Calculus uses graphs to describe change, limiting behaviour, and accumulated area.

Notation

\(f\)

a function

\(x\)

input variable

\(f(x)\)

output value of f at x

\(D_f\)

domain of f

\(\operatorname{range}(f)\)

set of outputs actually produced by f

The Core Method

To analyse a function, identify its domain, evaluate outputs, and track the shape of its graph.

Graph points

\[y=f(x)\]

The domain is restricted by operations such as division by zero and even roots of negative numbers. Intercepts come from setting one coordinate to zero: \(y\)-intercepts use \(x=0\), and \(x\)-intercepts solve \(f(x)=0\).

Worked Cases

Question

For \(f(x)=\frac{x+2}{x-3}\), find the domain and intercepts.

Answer

The denominator cannot be zero, so \(x-3\ne0\), giving \(x\ne3\). Thus the domain is all real \(x\) except \(3\). The \(y\)-intercept is \(f(0)=\frac{2}{-3}=-\frac23\). The \(x\)-intercept solves \(x+2=0\), so \(x=-2\).

Examples

Question

If \(f(x)=x^2-4\), identify where the graph crosses the axes.

Answer

For the \(y\)-intercept, set \(x=0\): \(f(0)=-4\). For the \(x\)-intercepts, solve \(x^2-4=0\). Factor: \((x-2)(x+2)=0\), so \(x=2\) or \(x=-2\).

Flashcards

Limit Laws