Domain & Range

Domain & Rangevideo

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 $\mathrm{f}\left(x\right)=...$ 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:

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

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

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

    • The domain is "x ≥ 0"

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

    • The domain is "all values of x"

  • $\mathrm{f}\left(x\right)=3x+2$ 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

  • $\mathrm{f}\left(x\right)=3x+2$ 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)

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

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

  •  $\mathrm{f}\left(x\right)=\sqrt{x-3}$ 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:

  • $\mathrm{f}\left(x\right)=3x+2$ 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)

  • $\mathrm{f}\left(x\right)=x^2$ 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