2.5 Finding Domains and Ranges of the Toolkit Functions
We will now return to our set of toolkit functions to determine the domain and range of each.
![Constant function f(x)=c.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_011.jpg)
![Identity function f(x)=x.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_012.jpg)
![Absolute function f(x)=|x|.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_013.jpg)
![Quadratic function f(x)=x^2.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_014.jpg)
![Cubic function f(x)-x^3.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_015.jpg)
![Reciprocal function f(x)=1/x.](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_016.jpg)
![Reciprocal squared function f(x)=1/x^2](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_017.jpg)
![Square root function f(x)=sqrt(x).](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_018.jpg)
![Cube root function f(x)=x^(1/3).](https://ecampusontario.pressbooks.pub/app/uploads/sites/2184/2021/11/CNX_Precalc_Figure_01_02_019.jpg)
How To
Given the formula for a function, determine the domain and range.
- Exclude from the domain any input values that result in division by zero.
- Exclude from the domain any input values that have non-real (or undefined) number outputs.
- Use the valid input values to determine the range of the output values.
- Look at the function graph and table values to confirm the actual function behaviour.
Example 1: Finding the Domain and Range Using Toolkit Functions
Find the domain and range of [latex]\text{}f\left(x\right)=2{x}^{3}-x[/latex].
Example 2: Finding Domain and Range
Find the domain and range of [latex]\text{}f\left(x\right)=\frac{2}{x+1}[/latex].
Example 3: Finding the Domain and Range
Example 4: Finding the Domain and Range
Find the domain and range of [latex]\text{}f\left(x\right)=-\sqrt{2-x}[/latex].
Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions