Domain & Range (Cambridge International IGCSE Mathematics)

Domain & range

How are functions related to graphs?

• Functions can be represented as graphs on x and y axes

  • The x-axis values are the inputs
  • The *y-*axis values are the outputs

• To see what graph to plot, replace $\mathrm{f} ( x ) =...$ with $y=...$

What is the domain?

• The domain of a function is the set of all inputs that the function is allowed to take

• Domains can be described in words

  • Domains must refer to x

    • not y or f(x)

  • You can use "not equal to" ≠ if needed

  • You can use inequality signs if needed

What are examples of domains?

• Examples of domains are below:

  • $\mathrm{f} ( x ) = \frac{1}{x}$ takes any x value except 0 (you cannot divide by 0)

    • The domain is "all values of x except 0", or simply "x ≠ 0"

  • $\mathrm{f} ( x ) = \sqrt{x}$ takes any x value that is not negative (you cannot take the square root of a negative)

    • The domain is "x ≥ 0"

  • $\mathrm{f} ( x ) = x^{2}$ takes any x value (negative x values are fine as inputs)

    • The domain is "all values of x"

  • $\mathrm{f} ( x ) = 3 x + 2$ takes any x value

    • The domain is "all values of x"

What are restricted domains and values excluded from domains?

• Some domains are restricted by choice

  • $\mathrm{f} ( x ) = 3 x + 2$ with the domain 0 < x < 5

    • This question wants to concentrate on that domain only (even though bigger domains exist)

• Some domains must exclude certain values (or sets of values)

  • $\mathrm{f} ( x ) = \frac{1}{( x - 1 ) ( x + 7 )}$ must exclude x = 1 and x = -7 from any domain

    • These two inputs make the function undefined (dividing by zero)

  • $\mathrm{f} ( x ) = \sqrt{x - 3}$ must exclude x < 3 from any domain

    • Any input in x < 3 leads to square-rooting a negative

What is the range?

• The range of a function is the set of all outputs that the function gives out

• Ranges can be described in words

  • Ranges must refer to f(x)

    • not x or y

  • You can use "not equal to" ≠ if needed

  • You can use inequality signs if needed

• Ranges depend on domains

• Examples of ranges are below:

  • $\mathrm{f} ( x ) = 3 x + 2$ with the domain x > 0

    • If x = 0 then f(0) = 3(0) + 2 = 2
    • The range is "f(x) > 2"
    • This is because if all inputs are greater than 0, all outputs will be greater than 2
    • This could be seen from a sketch or by substituting inputs of x > 0 into f(x)

  • $\mathrm{f} ( x ) = x^{2}$ with domain "all values of x"

    • The range is f(x) ≥ 0
    • This is because all values of x get squared (so no negative outputs are created)
    • Any negative value that goes in comes out positive
    • (0 goes in and comes out as 0² = 0)

How can I use graphs to find ranges?

• Ranges are easier if you know the shapes of different types of graphs

  • For example, the shapes of $y=\frac{1}{x}$, $y=x^{2}$, $y=x^{3}$, trig graphs, etc
  • They may also involve graph transformations

Exam Hint

• Sketching a function in an exam can help to "see" both the domain and range of that function

Worked Example Two functions are given by

$\mathrm{f} ( x ) = 10 - x \mathrm{g} ( x ) = \frac{1}{2 x - 1}$

(a) If the domain of the function $f$ is $2<x\leq4$, find the range.

Answer:

The domain is the set of inputs

Substitute x = 2 into f(x) to find its output

$\begin{array}{rcl} \mathrm{f} ( 2 ) & = 10 - 2 \\ = 8 \end{array}$

Substitute x = 4 into f(x) to find its output

$\begin{array}{rcl} \mathrm{f} ( 4 ) & = 10 - 4 \\ = 6 \end{array}$

Think of f(x) = 10 - x as a graph

the graph of $y=10-x$

This straight-line graph has a negative gradient *Between x = 2 and x = 4 the graph decreases from a height of 8 to a height of 6 *

Relate this to outputs

all outputs are between 6 and 8

Write down the range using f(x)

Remember that the inequality is "equal to" at x = 4, f(x) = 6 (this is the opposite order of "equal to" in the domain)

Final answer: The range is$6\leq f(x)<8$

(b) Write down the value of $x$ that must be excluded from the domain of function $g$.

Answer:

An input cannot cause the function to divide by zero Find out when "dividing by zero" would happen

$2x-1=0$

Solve to find this value of x (the one that must be excluded)

$2x=1\,x=\frac{1}{2}$

Final answer: $x=\frac{1}{2}$ must be excluded from the domain