The equation −3x − y² = −1 is not a function because it violates the definition of a function where each input should have a unique output.
To determine whether the equation −3x − y² = −1 represents a function, we need to check if there is a unique output (y) for every input (x) or if there are multiple outputs for the same input. In other words, we need to ensure that each x-value maps to only one y-value.
First, let's rearrange the equation to isolate y²:
−3x − y² = −1
y² = 3x - 1
Now, it's evident that for a single value of x, there could be two possible values of y due to the square root. This means that for some values of x, there would be multiple outputs (y-values). Therefore, the equation −3x − y² = −1 is not a function because it violates the definition of a function where each input should have a unique output.
For each value of x, we multiply it by −3, add 1, and then take the square root to get the y-value. For example, if x=−8, we have
We have that if x=−8, then y=5 or y=−5. In other words, a single value of x results in more than one value of y. Since some value of x corresponds to more than one value of y, the equation does not define a function.