Functions Organize Input and Output

A function is a rule assigning exactly one output to every permitted input. Function notation makes dependence visible and allows rules to be combined, compared, and inverted.

Learning objectives

You will evaluate functions, identify domains and ranges, use the vertical-line test, compose functions, and decide whether an inverse exists.

Function notation is substitution

If $f(x)=2x^2-3$, then

$f(-2)=2(-2)^2-3=5.$

For a compound input,

$f(x+h)=2(x+h)^2-3,$

not $f(x)+h$. The entire input replaces every occurrence of the function’s variable.

Representations answer different questions

A formula reveals algebraic structure. A table gives selected values. A graph reveals global shape and intercepts. A verbal rule supplies context. Translating among them is central to modeling.

The vertical-line test asks whether any input coordinate meets the graph more than once. If so, the relation assigns multiple outputs and is not a function of $x$.

Composition follows process order

If $g$ converts Celsius to Fahrenheit and $f$ converts Fahrenheit to a sensor voltage, then $f(g(x))$ is the complete pipeline. Generally $f\circ g\ne g\circ f$.

Inverses undo functions

An inverse satisfies $f^{-1}(f(x))=x$. It exists only when the original function is one-to-one on its chosen domain. Restricting $f(x)=x^2$ to $x\ge0$ gives inverse $f^{-1}(x)=\sqrt x$.

Check your understanding

Let $f(x)=3x-2$ and $g(x)=x^2$. Find $f(g(2))$ and $g(f(2))$. What does the comparison show?

Show the reasoning

$f(g(2))=f(4)=10$, while $g(f(2))=g(4)=16$. Composition order matters.