Functions
Domain and range
In this note
- 2.12Understand the domain (the inputs allowed) and the range (the outputs produced).
- 2.12Find values that must be excluded from a domain.
- 2.12Find the range of linear and quadratic functions, and read domain and range from asymptotes.
Before you start: Function notation and completing the square. Next: Asymptotes.
Formula bank
Inputs and outputs
The domain is the set of inputs a function is allowed to take. The range is the set of outputs it actually produces from those inputs. A function can't divide by zero or take the square root of a negative number, so those inputs are left out.
| Function | Domain | Why |
|---|---|---|
| \(f(x)=\dfrac5{x-2}\) | \(x\ne2\) | x = 2 would divide by zero |
| \(f(x)=\sqrt{x-3}\) | \(x\ge3\) | no square root of a negative |
| \(f(x)=4x-3\) | all real numbers | nothing goes wrong |
Try it yourself
Hint
Show answer
Answer −3
Finding the range
General methodHow to find the range
- Linear function on a domain: work out the outputs at the two ends of the domain; write smaller ≤ f(x) ≤ larger.
- Quadratic (all real x): complete the square, \((x+p)^{2}+q\). A square is never negative, so \(f(x)\ge q\).
- A list of inputs: work out each output and list each value once.
- Ends of the domain included (≤) → ends of the range included; strict (<) → strict.
Example \(m(x)=7-2x\), domain \(0\le x\le5\). Find the range.
Work out both ends; it is decreasing, so they swap round.
- \(m(0)=7\), \(m(5)=-3\)
Answer \(-3\le m(x)\le7\)
Example 2 \(f(x)=x^{2}+4x-3\). Find the range.
Complete the square; the square is never negative.
- \(f(x)=(x+2)^{2}-7\)
- \((x+2)^{2}\ge0\)
Answer \(f(x)\ge-7\)
Try it yourself
Show answer
Answer \(1\le g(x)\le7\)
Show answer
Answer 3
Reading domain and range from a graph
For a curve with asymptotes, the domain and range come straight from the asymptote lines. \(y=\dfrac a{x-b}+c\) has a vertical asymptote \(x=b\) (that x is never allowed) and a horizontal asymptote \(y=c\) (that y is never reached). So the domain is \(x\ne b\) and the range is \(y\ne c\). For \(y=2^{x}+1\) the curve stays above \(y=1\): range \(y>1\).
| Function | Asymptotes | Domain | Range |
|---|---|---|---|
| \(y=\dfrac3{x-2}+1\) | \(x=2\), \(y=1\) | \(x\ne2\) | \(y\ne1\) |
| \(y=\dfrac1{x}\) | \(x=0\), \(y=0\) | \(x\ne0\) | \(y\ne0\) |
| \(y=2^{x}+1\) | \(y=1\) | all x | \(y>1\) |
Ways they test it
Type 1 Range of a linear function
"\(k(x)=5x-4\), domain \(-2\le x\le3\). Find the range." (2 marks)
Spot it a domain with two ends.
- Work out both ends.
- Smaller first, same type of signs.
For example \(k(-2)=-14\), \(k(3)=11\)
Answer \(-14\le k(x)\le11\)
Type 2 Excluded value
"Which value of x must be excluded from the domain of \(\dfrac4{2x+6}\)?" (1 mark)
Spot it x in a denominator.
- Set the denominator = 0.
- Solve.
For example \(2x+6=0\)
Answer \(x=-3\)
Type 3 Range of a quadratic
"Write \(x^{2}-6x+11\) as \((x-a)^{2}+b\). Hence state the range." (3 marks)
Spot it 'hence' after completing the square.
- Complete the square.
- Range: f(x) ≥ b.
For example \((x-3)^{2}+2\)
Answer \(f(x)\ge2\)
Type 4 Mapping a list
"h(x) = x² + 1 with domain {−2, −1, 0, 1, 2}. List the range." (1 mark)
Spot it a set of inputs in curly brackets.
- Work out each output.
- List each value once.
For example outputs 5, 2, 1, 2, 5
Answer {1, 2, 5}
Mistakes to avoid
- ✗ Range = the domain✓ work out the outputsDifferent sets.
- ✗ \(7\le m(x)\le-3\)✓ \(-3\le m(x)\le7\)Smaller value first.
- ✗ Range of x² + 3 is x ≥ 3✓ f(x) ≥ 3A range is about the outputs.
Summary
- Domain = inputs allowed; range = outputs produced.
- Exclude values that make a denominator zero or a square root negative.
- Linear: the two ends. Quadratic: complete the square, \(f(x)\ge q\).
- Asymptotes give the domain (x ≠ …) and range (y ≠ …) straight away.
If you see …
- "find the range" of a linear functionwork out the outputs at both ends of the domain.
- a quadratic with no domain givencomplete the square: range ≥ the minimum.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer \(-2\le g(x)\le13\)
Show answer
Answer −5
Show answer
Answer 1, 2, 5
Medium
Guide me
What is it asking? an inequality
What do we know? work out k(−2) and k(3)
Plan smaller first
Show working
- \(k(-2)=-14\)
- \(k(3)=11\)
Answer \(-14\le k(x)\le11\)
Guide me
What is it asking? an inequality
What do we know? decreasing
Plan m(−1) = 9, m(3) = 1
Show working
- \(m(-1)=9\)
- \(m(3)=1\)
Answer \(1\le m(x)\le9\)
Guide me
What is it asking? the minimum output
What do we know? \(x^{2}\ge0\)
Plan so \(f(x)\ge-4\)
Show working
- \(x^{2}\ge0\Rightarrow x^{2}-4\ge-4\)
Answer −4
Hard
Guide me
What is it asking? a, b, minimum
What do we know? half of −6 is −3
Plan subtract 9, add 11
Show working
- \((x-3)^{2}-9+11=(x-3)^{2}+2\)
- range \(f(x)\ge2\)
Answer \((x-3)^{2}+2\); minimum 2
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \((x-3)^{2}+2\) | B2 | ||
| \(f(x)\ge2\) | B1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? the minimum
What do we know? \((x+2)^{2}-7\)
Plan the square is never negative
Show working
- \((x+2)^{2}-4-3\)
- \(f(x)\ge-7\)
Answer −7
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \((x+2)^{2}-7\) | B2 | ||
| \(f(x)\ge-7\) | B1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? two values
What do we know? vertical asymptote where \(x-2=0\)
Plan horizontal asymptote at \(y=1\)
Show working
- \(x-2=0\Rightarrow x=2\)
- \(\dfrac3{x-2}\) is never 0, so \(y\) is never 1
Answer \(x\ne2\), \(y\ne1\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(x\ne2\) | B1 | ||
| \(y\ne1\) | B1 | ||
| Total: 2 marks | |||
Can I do this?
- I can explain domain and range and find excluded values.
- I can find the range of linear and quadratic functions.
- I can read domain and range from the asymptotes of a graph.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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