Teach Me Maths

Functions - Domain and Range

How to use function notation and find allowed inputs and possible outputs, including restricted domains and endpoint conditions.

Inputs and outputs

f(x) means the output when x is put into the function f. It does not mean f multiplied by x. Replace every occurrence of the input, using brackets for negative numbers.

The domain is the set of allowed inputs. The range is the set of outputs actually reached. Use x for input conditions and f(x), or y, for output conditions.

Worked example 1

For f(x)=2x^2-5x+3, find f(-2)

  1. f(-2)=2(-2)^2-5(-2)+3. The square applies to the whole input.

  2. Your turn. What is (-2)^2?

    (-2)^2=4

  3. Answer: f(-2)=8+10+3=21

Find the allowed inputs

Start with any domain stated in the question. For real-valued functions, a denominator cannot be zero and the expression under a square root must be non-negative. Apply all restrictions together.

Worked example 2

Find the largest real domain of h(x)=\frac{\sqrt{2x+6}}{x-1}

  1. The square root requires 2x+6\ge0

  2. Your turn. Solve 2x+6\ge0. What is the lower bound for x?

    x\ge-3

  3. The denominator requires x-1\ne0, so exclude x=1
  4. Answer: all real x with x\ge-3 and x\ne1. The endpoint -3 is allowed because its denominator is not zero.

Check which endpoints are reached

For a strictly increasing or decreasing function on an interval, compare the endpoint outputs. Keep track of which input endpoints are included. A strict inequality stays strict at the corresponding output endpoint.

A quadratic may turn inside the domain, so checking only its endpoints can miss its smallest or largest output.

Worked example 3

Find the range of f(x)=3-2x for -2\le x<4

  1. The function decreases as x increases. At x=-2, the output is 3-2(-2)

  2. Your turn. Work out this largest output.

    f(-2)=7

  3. As x approaches 4 from below, f(x) approaches -5. The input 4 is excluded, so -5 is not reached.
  4. Answer: -5<f(x)\le7

Explore a restricted quadratic

Try it: move the right endpoint past x=1. Once the turning point is included, the minimum remains -2. Increasing the domain need not change both ends of the range.

Worked example 4

Find the range of f(x)=(x-2)^2+1 for -1\le x\le4

  1. The turning point is at x=2, which lies in the domain. There the squared bracket is zero.

  2. Your turn. What is the minimum output?

    f(2)=1

  3. At the endpoints, f(-1)=(-3)^2+1=10 and f(4)=2^2+1=5. The largest output is 10
  4. Answer: 1\le f(x)\le10. Both bounds are reached.

Worked example 5

Find the range of g(x)=2x^3-1 for -1<x\le2

  1. The cubic x^3 is increasing for all real x. Multiplying by positive 2 and shifting down preserves that order.
  2. The lower endpoint would give g(-1)=-3, but x=-1 is excluded.

  3. Your turn. Find g(2)=2(2)^3-1

    g(2)=15

  4. Answer: -3<g(x)\le15

Worked example 6

Find the range of f(x)=3\times2^x+1 for all real x

  1. 2^x>0 for every real x, so 3\times2^x is always positive. It approaches zero as x decreases without bound.

  2. Your turn. What output value is approached after adding 1?

    The output approaches 1 but never reaches it.

  3. There is no upper bound, because 2^x grows without bound as x increases.
  4. Answer: f(x)>1, not f(x)\ge1

Common mistakes

  • Giving inputs as the range. The domain describes inputs; the range describes outputs.
  • Substituting a negative input without brackets. Bracket the whole input before applying powers or other operations.
  • Allowing a zero denominator. Exclude every input that makes a denominator zero.
  • Checking only the endpoints. Also check turning points inside the domain when finding the range.
  • Including an excluded boundary value. Preserve strict inequalities unless that output is reached elsewhere in the domain.
  • Including an unreached asymptote value. Include an output only if an allowed input produces it.

Now try it: Functions - Domain and Range practice questions

More on this topic: Functions - Domain and Range worksheet with full solutions