An equation defines a function if, for every input value, there is only one output value (or no output value).
Here's how to determine if an equation defines a function, incorporating the provided reference:
The key concept is understanding the relationship between inputs and outputs. According to the reference, a function will only have one or zero outputs for any input. If you find even a single input that leads to more than one output, then the equation does not define a function.
To determine whether an equation represents a function, you can use the following methods:
- Vertical Line Test (for graphs): If any vertical line intersects the graph of the equation more than once, the equation does not represent a function. This is a visual way to check if one x-value (input) maps to more than one y-value (output).
- Solve for y: If you can solve the equation for y and end up with only one possible y-value for each x-value, then the equation likely defines a function. If solving for y introduces a ± (plus or minus) sign or absolute value that results in two different y values for a single x-value, it's an indicator that it is not a function.
- Consider the equation's form: Certain forms immediately indicate non-functions. For example,
x = y²is a common example of an equation that does not define y as a function of x because for any positive value of x, there are two possible values of y (one positive and one negative).
Examples
Here are a few examples to illustrate the concept:
| Equation | Function? | Explanation |
|---|---|---|
| y = x + 2 | Yes | For every x-value, there is only one y-value. |
| y = x2 | Yes | For every x-value, there is only one y-value. |
| x = y2 | No | For x = 4, y could be 2 or -2. Since one x value maps to two y values, it's not a function. This violates the function definition of one or zero outputs for any input. |
| y2 = x2 + 1 | No | Solving for y yields y = ±√(x2 + 1). For any x, there are two y values (positive and negative root), so it is not a function. It has more than one output for an input. |
| y = √x | Yes | Though the square root has both a positive and negative result, √x as a function is defined to be the principal square root, or positive result. So for every x, there is only one y. |
Therefore, carefully analyze the equation to ensure that each input (usually x) corresponds to only one output (usually y). If this condition is met, the equation defines a function.
