Putting it all together...
When an equation is a function, each value in the domain corresponds with exactly one value in the range. If the equation has both the x and y variables raised to the first power, and there are no square root signs, then the equation is linear. It is a function. All linear functions have a domain and range of all real numbers. When an equation is a function, you can replace the output variable with function notation.
Function notation indicates that the output variable is a function of the input variable. You can use this notation to evaluate functions. For example:
Evaluate \( f\left( x \right) = 2x + 7 \) for \( f( - 4) \). What are the domain and range of this function?
| Evaluate \( f\left( x \right) = 2x + 7 \) for \( f( - 4) \) means to substitute –4 for x on the right-hand side of the function. | \( f\left( - 4 \right) = 2\left( - 4 \right) + 7 \) \( f\left( - 4 \right) = - 8 + 7 \) \( f\left( - 4 \right) = - 1 \) |
| Find the domain and range of this function. | Since both the input and output variables are raised to the first power, this function is linear. The domain and range are all real numbers. |
Use the activity below to practice finding functions and using function notation. Look at the equation in the far left column. Evaluate it for \( x = 3 \). Use your answer to help you determine if the equation is a function. If the equation is a function, write it using function notation. Then click on the equation to check your answer.
| Equation | Value Obtained from x = 3 | Is this a Function? | Function Notation |
|---|---|---|---|
\( y = \sqrt{3 - 3} \) \( y = 0 \) |
Yes |
\( f\left( x \right) = \sqrt{x - 3} \) |
|
\( y = 3 - 9 \) \( y = - 6 \) |
Yes |
\( f\left( x \right) = x - 9 \) |
|
\( \left( 3 \right)^{2} + y^{2} = 100 \) \( y = \pm \sqrt{91} \) |
No |
This is not a function. |
|
\( y = {(3)}^{2} \) \( y = 9 \) |
Yes |
\( f\left( x \right) = x^{2} \) |
|
\( y = \left| \left( 3 \right) + 8 \right| - 7 \) \( y = 4 \) |
Yes |
\( f(x) = \left| x + 8 \right| - 7 \) |
|
\( x = y^{2} - 6 \) \( 3 = y^{2} - 6 \) \( \pm 3 = y \) |
No |
This is not a function. |
|
\( y = 35\left( 3 \right) - 50 \) \( y = 55 \) |
Yes |
\( f(x) = 35x - 50 \) |
Question
What are the domain and range of \( f(x) = 35x - 50 \)? Explain.
Both the input and output variables are raised to the first power, and there are no square root symbols. This function is linear, so its domain and range are all real numbers.
