What can go in, and what can come out?

A relation needs exactly one output per allowed input to be a function

The source represents $y=x+2$ in four ways: a table $(0,2),(1,3),(2,4),(3,5)$, arrows between input/output sets, a straight line with intercepts $(-2,0),(0,2)$, and the equation.

The table/arrows give selected pairs; the drawn full line describes the real-input extension.

All functions are relations, but some relations assign two outputs to one input.

In the source comparison, an extra arrow from input 0 to output 3 makes that relation fail the function definition, while all four original arrows remain. The nested-set diagram places functions inside relations.

For the source square mapping, $1\mapsto1$, $2\mapsto4$, $3\mapsto9$, $4\mapsto16$. Thus $D_f=\{1,2,3,4\}$ and $R_f=\{1,4,9,16\}$; one possible rule is $f(x)=x^2$. Present domains and ranges as sets or intervals. The notation $f:x\mapsto y$ or $f(x)=y$ names an output; $f$ names the function. A function machine takes x as input and emits f(x).

Give both the rule and its domain: $g:x\mapsto x^2+5$, $x<0$, and $g(x)=x^2+5$, $x<0$, express the same definition. The source says an omitted domain means all real numbers. Qualify that convention for real-valued formulas: the rule must still be defined, so a logarithm requires a positive argument and a denominator cannot be zero.

The vertical-line test checks whether each allowed input has at most one graph point; the relation must also give an output for every input in its declared domain. Do not confuse this with the horizontal-line test for an inverse. Example 2(b) is $\pm\sqrt x$, with both branches, not $\pm x$; the radical is visible in the PDFs although lost in text extraction. The Teacher’s line x=2 demonstrates the failure.

Read the range on the actual domain

Find every output reached by an allowed input. Work in this order:

1. Mark the allowed input intervals and any holes where the rule is undefined.
2. Check included endpoints and turning points inside those intervals.
3. Find the limits at excluded endpoints and infinity. A limit alone does not show that an output is reached.
4. Combine the reached outputs. Include a boundary only if some allowed input attains it; keep any gaps.

For a continuous curve on an interval, the outputs between its extreme values are reached. For separate pieces, check each piece before combining their ranges.

Use a filled circle for an included endpoint and a hollow circle for an excluded endpoint. An open input endpoint does not automatically make the range endpoint open: another allowed input may reach the same output.

Apply the method to the source examples

In Example 3 the first curve decreases across an interval with one endpoint excluded, whereas the second has an interior minimum despite an excluded left boundary. Rounded source values such as −5.39 and 0.131 describe approximate endpoints; the working preserves exact/implicit values separately. Its complete nine-row calculator procedure sits beside the exercise, with original screens retained in evidence. Removing a temporary interval restriction never removes the natural logarithm domain.

Even and odd functions preserve the whole domain

For an even function, $f(-x)=f(x)$: $(a,b)$ pairs with $(-a,b)$, reflecting in the y-axis. The source examples are $\cos x$ and $x^2$.

For an odd function, $f(-x)=-f(x)$: $(a,b)$ pairs with $(-a,-b)$, a half-turn (angle $\pi$) about the origin. The examples are $\sin x$ and $x^3$.

Independently, the domain must be symmetric under $x\mapsto-x$ for these whole-function tests. Check domain restrictions as well as formulas.

The source contains stray axis labels in these illustrations; the reconstructions use x horizontally and y vertically while retaining the intended curves.

Select a piece using its input condition

A piecewise function uses specified rules on intervals. The source recalls $|x|=x$ for $x\ge0$ and $|x|=-x$ for $x<0$, giving the V-shaped graph.

In Example11, test each endpoint against its own inequality before finding the union of output sets.

An open point on one piece may still be reached elsewhere; check all pieces. Here neither piece reaches4.

Its four-stage calculator procedure illustrates how to enter two pieces and inspect them with ZBox, but exact endpoint coordinates require algebra.

Repeat one interval, then clip the requested graph

Periodicity means a nonzero translation preserves the function: $f(x+a)=f(x)$ for every input in the repeated domain. The source cosine illustration marks four consecutive intervals of length $2\pi$.

A positive period need not be the least positive period.

Example12 uses period4; reduce each input to $(0,4]$ before selecting a branch. Its periodic extension defines the values at0 and negative inputs as well.

Label all endpoints and joins in the requested sketch, preserving its open left endpoint and closed right endpoint.

The preceding source remark calls $f^{-1}f$ and $ff^{-1}$ identity functions: each maps an accepted input to itself. Example10 in the composition lesson retains the important distinction between their domains. First mark the allowed input set. Check turning points inside it, included endpoints, and limits at excluded endpoints or infinity. Include an output boundary only if an allowed input actually reaches it.