Functions and Domains

• In $\mathrm{f}(x)=x+2$, $x$ is the input and $x+2$ is the output. Put in $5$, get $\mathrm{f}(5)=7$.
• $\mathrm{f}:x\mapsto x+2$ says “f maps x to x + 2”. The arrow gives the rule; it is not an equality sign.
• $x\in\mathbb{R}$ means that $x$ may be any real number: any ordinary number on the number line.

On a graph, use the vertical-line test. If one vertical line meets the graph twice, the same $x$ has two outputs, so the graph does not represent a function.

• The domain is the set of allowed inputs: how far the graph extends left to right.
• The range is the set of outputs actually produced: the heights the graph reaches.

Domain is input; range is output. A function may rise, fall, turn or repeat an output. It only fails to be a function when one input has more than one output.

If no domain is stated, use the largest set of real inputs for which the formula gives a real output. This is the natural domain.

• A polynomial such as $x^2-3x+1$ accepts every real input.
• A denominator cannot be zero. For $\mathrm{f}(x)=\dfrac{1}{x-4}$, exclude $x=4$.
• Inside a real square root must be at least zero. For $\mathrm{g}(x)=\sqrt{7-2x}$, solve $7-2x\ge0$, so $x\le\tfrac72$.
• Apply all restrictions together. For $\mathrm{h}(x)=\dfrac{\sqrt{7-2x}}{x+1}$, the domain is $x\le\tfrac72$ with $x\ne-1$.