Porisma

What can go in, and what can come out?

Read domains and ranges as sets, including turning points and values that are only approached.

← Back to lesson · Complete chapter reading and print view · Open reference notes separately

By the end, you can

  • Distinguish allowed inputs from reached outputs
  • Find a range using endpoints and turning points
  • Explain open and closed interval boundaries

about 20 minutes for the lesson · practise at your own pace.

The central idea

The domain tells us which inputs are allowed. The range contains exactly the outputs reached from those inputs. For f(x)=x2−1f(x)=x^2-1 with x∈Rx\in\mathbb R, the domain is R\mathbb R and the range is [−1,∞)[-1,\infty). Squaring never gives a negative value, and f(0)=−1f(0)=-1 reaches the lower boundary.

Four representations of a relation

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Source y=x+2 shown as table 0→2,1→3,2→4,3→5, corresponding set arrows, equation and full line with intercepts −2 and 2. Functions form a subset of relations.
Source y=x+2 shown as table 0→2,1→3,2→4,3→5, corresponding set arrows, equation and full line with intercepts −2 and 2. Functions form a subset of relations.
Source y=x+2 shown as table 0→2,1→3,2→4,3→5, corresponding set arrows, equation and full line with intercepts −2 and 2. Functions form a subset of relations.
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One output per input, and the square mapping

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Source valid mapping 0→2,1→3,2→4,3→5 versus invalid relation with an additional 0→3 arrow. Square mapping 1→1,2→4,3→9,4→16 has domain{1,2,3,4}, range{1,4,9,16}. Function machine takes x to f(x).
Source valid mapping 0→2,1→3,2→4,3→5 versus invalid relation with an additional 0→3 arrow. Square mapping 1→1,2→4,3→9,4→16 has domain{1,2,3,4}, range{1,4,9,16}. Function machine takes x to f(x).
Source valid mapping 0→2,1→3,2→4,3→5 versus invalid relation with an additional 0→3 arrow. Square mapping 1→1,2→4,3→9,4→16 has domain{1,2,3,4}, range{1,4,9,16}. Function machine takes x to f(x).
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Even and odd source curves

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Cosine and x² reflect in the y-axis; sine and x³ have half-turn symmetry about the origin. Horizontal axis x and vertical axis y correct stray source axis labels.
Cosine and x² reflect in the y-axis; sine and x³ have half-turn symmetry about the origin. Horizontal axis x and vertical axis y correct stray source axis labels.
Cosine and x² reflect in the y-axis; sine and x³ have half-turn symmetry about the origin. Horizontal axis x and vertical axis y correct stray source axis labels.
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Modulus selects one of two rules

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V-shaped y=absolute x consists of y=−x for x<0 and y=x for x≥0, joined at included origin.
V-shaped y=absolute x consists of y=−x for x<0 and y=x for x≥0, joined at included origin.
V-shaped y=absolute x consists of y=−x for x<0 and y=x for x≥0, joined at included origin.
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Cosine repeats after two pi

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Cosine from −4π to 4π with four adjacent peak-to-peak intervals each labelled 2π.
Cosine from −4π to 4π with four adjacent peak-to-peak intervals each labelled 2π.
Cosine from −4π to 4π with four adjacent peak-to-peak intervals each labelled 2π.
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Explore

Move the left boundary. Opening it excludes that endpoint. When the boundary passes the turning point, the minimum changes.

Where is the smallest output?

Move the boundary past the vertex and compare the input and output intervals.

Independent warm-up: f(x) = x² − 1, restricted to x ≥ k or x > k.

Left boundary k-1

Closed endpoint: the boundary is included unless the original domain excludes it.

Try an example

Proposed input domain

[-1, ∞)

Proposed input domain-303

Retained inputs

[-1, ∞)

Retained inputs-303

Reached outputs

[-1, ∞)

Reached outputs-1.53.258

Original allowed inputs

(−∞, ∞)

These exclusions stay in force when a boundary changes. Arrows indicate continuation beyond the drawing window.

The allowed inputs contain 0. The minimum output −1 is reached, regardless of the left boundary being open or closed.

The graph shows a finite window; arrows and interval notation describe the full set. Decimal endpoints are rounded.

Worked example

For f(x)=x2−1f(x)=x^2-1 on x>−1x>-1, the input boundary −1-1 is excluded, but x=0x=0 is still allowed and reaches −1-1. The range is therefore [−1,∞)[-1,\infty). For x>1x>1, the graph is increasing throughout the allowed domain and its lower output limit is 0, never reached. Now the range is (0,∞)(0,\infty).

A bracket in the domain does not transfer directly into a bracket in the range. Ask whether the output is reached anywhere in the allowed domain.

Check your understanding

Find the range of p(x)=(x−3)2+2p(x)=(x-3)^2+2 on x>1x>1. Does excluding the input 1 exclude the minimum output?

Practice questions

Practice section 1 · Questions 1–6

Translate all seven interval rows · Substitute a complete input expression · +4 more

Practice section 2 · Questions 7–11

Repeat the defining interval without losing endpoints · Three ranges, three endpoint decisions · +3 more

More practice

Practise reading ranges before using them to decide whether an inverse or a composite exists.

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