3.1 Function notation and transforms

A function is a rule that gives one output for each allowed input. Function notation lets you state that rule once and ask many questions about it. A graph shows all of those input-output pairs at once. This language is used throughout later modules, from population models to classifier scores.

This is foundational algebra with an intermediate visual step. If substitution into a formula is familiar, start with OpenStax's function-notation lesson. Its graph-transformation lesson helps when it is hard to tell whether a change acts on the input or the output.

A basic idea: one input, one output

If $f(x)=2x+3$, then $f(4)=2(4)+3=11$. The expression $f(4)$ asks for the output when the input is $4$; it does not mean $f$ multiplied by $4$. A function's domain is the set of inputs it allows. For $g(x)=\sqrt{x-1}$ over the real numbers, the domain is $x\ge1$, because a negative number under a square root has no real value.

A graph places each allowed pair $(x,f(x))$ on coordinate axes. A vertical line should meet the graph of a function at most once: one input cannot have two different outputs.

A quick tour of transformations

Start with a known graph $y=f(x)$. The following rules describe a new graph:

The horizontal shift can feel backwards: $f(x-2)$ moves a graph right by $2$, because the new input $x=2$ supplies the old input $0$. If you combine changes, substitute carefully. For example, $2f(x-2)+1$ moves the original graph right $2$, doubles its heights, and then moves it up $1$.

For a simple picture, take $f(x)=|x|$. The new rule $g(x)=|x-2|+1$ moves its V-shaped graph right $2$ and up $1$.

Application 1: reading a cost rule

A print shop charges a setup cost of $20$ units and $3$ units per item. Let $C(n)=20+3n$, where $n$ is a nonnegative whole number. What is the cost of $7$ items, and how many items cost $50$ units?

1. Evaluate the rule. $C(7)=20+3(7)=41$ units.
2. Solve for the input. Set $20+3n=50$, so $3n=30$ and $n=10$.
3. Check the domain. Both $7$ and $10$ are allowed whole-number counts. A formula can produce $C(-2)=14$, but negative items are outside this problem's domain.

The function notation keeps the setup cost and per-item cost in one reusable rule.

Application 2: identify a moved graph

The graph of $y=|x|$ has its corner at $(0,0)$. A new V-shaped graph has its corner at $(2,1)$ and the same steepness. What is its formula, and what does it output at $x=5$?

1. Move the corner horizontally. Right by $2$ gives $|x-2|$.
2. Move it vertically. Up by $1$ gives $g(x)=|x-2|+1$.
3. Evaluate and check. $g(5)=|5-2|+1=4$. The point $(5,4)$ is three units right and three units above the new corner, matching the original V's steepness.

Identifying a graph is easier when you first locate an anchor point such as a corner, vertex, or intercept.