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.

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.

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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

11 examples and practice questions, in 2 sections.

More practice

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

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