Math101 · Independent Ontario learning library
Math101Introduction to Functions
A function assigns exactly one output to each allowed input and provides a precise language for relationships, graphs, tables, and models.
A function is a relationship in which every allowed input has exactly one output.
The machine idea
Imagine a machine that accepts an input, applies one dependable rule, and produces an output. If the rule is “triple, then subtract two,” input $5$ produces $13$.
Different inputs may share an output, but one input cannot produce two different outputs within the same function. That single-output rule is the defining feature.
Function notation
$f(x)$ means “the output of function $f$ at input $x$.” It does not mean $f$ multiplied by $x$.
If $f(x)=3x-2$, then
The letter naming a function can change. $g(t)$ may describe height at time $t$; notation helps identify both the rule and input variable.
Domain and range
The domain is the set of allowed inputs. The range is the set of outputs actually produced.
Context and algebra can restrict the domain. For $f(x)=1/(x-4)$, $x=4$ is excluded because it would divide by zero. In a model of ticket sales, negative or fractional ticket counts may be excluded even if the formula accepts them algebraically.
Evaluating and solving
Evaluating supplies an input and asks for the output. Solving $f(x)=10$ supplies an output and asks which inputs produce it.
Common mistakes
Reading $f(x)$ as multiplication. It names an output.
Reversing domain and range. Domain is input; range is output.
Believing repeated outputs are forbidden. Repeated inputs with different outputs are forbidden.
Applying the horizontal-line test. The vertical-line test identifies functions; horizontal lines test one-to-one behaviour.
Quick self-check
- Does each input have exactly one output?
- What restrictions belong to the domain?
- Am I evaluating an input or solving for an input?
- Which representation best reveals the feature I need?
