Home on the Range

Learning Focus

Describe the domain and range of a sequence and see why they are different from those of a continuous function.

Big idea

A sequence $f(n)$ is a function whose domain is the term numbers: $n=1,2,3,\dots$ (positive integers, or a finite list if the sequence stops).

The range is the set of output values (the terms themselves).

• The graph is a set of separate points, not a connected line.
• The domain is discrete: there is no term $2.5$.
• The range is also a list of separate values.

Worked example

Let $f(n)=2n+1$ for $n=1,2,3,\dots$

• Domain: $\{1,2,3,\dots\}$
• Range: $\{3,5,7,9,\dots\}$ (odd numbers starting at 3)

If we only look at the first 5 terms, the domain is $\{1,2,3,4,5\}$ and the range is $\{3,5,7,9,11\}$.

A line $y=2x+1$ would allow every real $x$; the sequence does not.

Takeaways

• Domain of a sequence = term numbers (positive integers).
• Range = the actual terms produced.
• Sequence graphs are dots; the domain restriction is what makes them different from lines.