A function is a rule that assigns exactly one output to each allowed input. In high school, functions help you connect equations, tables, and graphs.

What is a function?

If \(f\) is a function, then \(f(x)\) means “the output when input is \(x\).” Example:

\[ f(x)=2x+3. \]

Then \(f(1)=5\), \(f(4)=11\), and \(f(-2)=-1\).

Domain and range

The domain is the set of allowed inputs. The range is the set of outputs produced from that domain.

Example 1. \(f(x)=\frac{1}{x-2}\)

• Domain: all real \(x\) except \(x=2\).
• Range: all real \(y\) except \(y=0\).

Example 2. \(g(x)=\sqrt{x+1}\)

• Domain: \(x \ge -1\) (square root needs non-negative inside).
• Range: \(y \ge 0\).

Function vs relation

Not every relation is a function. If one input gives two outputs, it is not a function.

On a graph, use the vertical line test: if any vertical line crosses the curve more than once, it is not a function.

Graph basics

To sketch a graph:

1. Choose key \(x\)-values and make a value table.
2. Plot points \((x,f(x))\) on labeled axes.
3. Connect appropriately (line, curve, or separate branches).
4. Identify intercepts and turning features.

Important function families

Linear: \(y=mx+c\)

Straight line with slope \(m\) and \(y\)-intercept \(c\). A larger \(|m|\) means steeper line.

Quadratic: \(y=ax^2+bx+c\)

Parabola. If \(a>0\), opens upward; if \(a<0\), opens downward.

Absolute value: \(y=|x|\)

V-shaped graph with vertex at the origin before transformations.

Rational: \(y=\frac{1}{x}\)

Hyperbola with asymptotes \(x=0\) and \(y=0\). Graph has two separate branches.

Transformations

Start from \(y=f(x)\):

• \(y=f(x)+k\): shift up by \(k\).
• \(y=f(x-h)\): shift right by \(h\).
• \(y=af(x)\): vertical stretch/compression by factor \(|a|\), and reflect if \(a<0\).
• \(y=f(bx)\): horizontal scale by factor \(\frac{1}{|b|}\), and reflect in \(y\)-axis if \(b<0\).

Example. From \(y=x^2\) to \(y=(x-3)^2+2\): move right 3, up 2.

Intercepts and meaning

• \(y\)-intercept: set \(x=0\).
• \(x\)-intercepts: solve \(f(x)=0\).

In applications, intercepts often represent starting value and break-even/zero-output points.

Piecewise functions

Some rules change in different intervals. Example:

\[ f(x)= \begin{cases} x+2, & x<1\\ 3, & x\ge 1 \end{cases} \]

Graph each rule only on its stated interval, and mark endpoints correctly (open/closed points).

Exam strategy

• State domain restrictions before graphing.
• Use a neat value table near turning points or asymptotes.
• Label key coordinates: intercepts, vertex, asymptote equations.
• Check whether a requested equation is function form or relation form.