In a function, each value in the domain corresponds with exactly one value in the range. Usually the domain refers to x-values, and the range refers to y-values.
All the examples you have seen so far have contained coordinate pairs that produce points on a graph. But more often, you will be working with equations that can produce an infinite number of values and that you can graph using a line or a curve.
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You will need to be able to determine if these equations are functions, and you will need to be able to name their domain and range. Click each tab to see an example.
Evaluate \( y^{2} = x \) for \( x = 64 \). Is this equation a function?
Reminder
To evaluate means to substitute the given value for the named variable and then solve.
| Substitute \( x = 64 \) and solve. Remember that when you find the square root of a number, it produces both a positive and a negative answer. |
\( y^{2} = x \) \( y^{2} = 64 \) \( y = \pm 8 \) |
| Decide if this equation is a function. | Since the value \( x = 64 \) is paired with both –8 and 8, this equation is not a function. |
Evaluate \( y = 15x + 3 \) for \( x = - 1,\ 0,\ 2 \). Is this equation a function?
| You are asked to substitute each of three values into the given equation. | For \( x = - 1 \) \( y = 15( - 1) + 3 \) \( y = - 15 + 3 = - 12 \) For \( x = 0 \) \( y = 15(0) + 3 \) \( y = 0 + 3 = 3 \) For \( x = 2 \) \( y = 15(2) + 3 \) \( y = 30 + 3 = 33 \) |
| Decide if this equation is a function. | None of the x-values produced more than a single y-value. This indicates that this equation is likely a function. |
| In the first example, you were able to easily see that the equation \( y^{2} = x \) is not a function. The value x = 64 is paired with both –8 and 8. |
In the second example, \( y = 15x + 3 \), the values you substituted each produced only one y-value. This indicates that the equation may be a function. |
Question
What do you notice about the equations \( y^{2} = x \) and \( y = 15x + 3 \)?
The equation \( y^{2} = x \) has a square on the y. The equation \( y = 15x + 3 \) does not have a square on the y.
As a general rule, if an equation has an output variable (such as y) that is raised to an even power, 2, 4, 6, etc., then that equation is NOT a function.
If both the x and y variables are raised to the first power and there are no square root signs, then the equation is linear, and 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.
For Example
Write \( y = 15x + 3 \) using function notation.
| Function notation utilizes a letter, a set of parentheses, and the input variable. | For this equation, the input variable is x. You can choose your letter, but usually the letter f is used since the word function starts with the letter f. Using function notation, \( f\left( x \right) = 15x + 3 \). Other commonly used letters for functions include g and h: \( g\left( x \right) = 15x + 3 \) \( h\left( x \right) = 15x + 3 \) |
Question
Is the equation \( x^{2} + y^{2} = 50 \) a function? Explain.
HINT: Evaluate for \( x = 5 \).
No, this equation is not a function. If you evaluate the function for \( x = 5 \), then \( y = \pm 5 \). Since one x-value is matched with two y-values, this equation is not a function. You can also determine that this equation is not a function from the fact that the y-variable is raised to an even power.
