State if the inverse of y=x^2-4 is a function.
So, I got y= squareroot of x+4
I'm don't think it's a function because of even roots, but I'm not sure
Woahhh, let's back up here, how did you get that inverse?
How do you find the inverse of a function?
- you interchange the x and y variables.
so the inverse of y = x^2 + 4 is
x = y^2 + 4
- now solve this for y
y^2 = x - 4
y = ±√(x-4)
now let x = 8 , (or some other value of x you feel like)
then y = 2 or -2
so we have two points (8,2) and (8,-2)
This violates the rule of a function in that for every x I choose there can be one and only one value of y
You might also investigate the vertical line test to see if a relation is a function.
I solved for y and got y = ±√(x+4) for the inverse, the given equation was
y=x^2-4 not y = x^2 + 4
but your explanation helped understand why the inverse is not a function, thank you!
To determine if the inverse of a function is also a function, we need to check if the original function passes the horizontal line test. The horizontal line test states that for a function to be considered a function, any horizontal line drawn across the graph should intersect the function at most once.
Let's analyze the original function, y = x^2 - 4. To find its inverse, we first need to swap the positions of x and y, giving us x = y^2 - 4.
Now, we can solve the equation for y. Starting with x = y^2 - 4:
x = y^2 - 4
x + 4 = y^2
y^2 = x + 4
Next, we take the square root of both sides:
√(y^2) = √(x + 4)
y = ±√(x + 4)
When finding the inverse of a function, we include both the positive and negative square root because the original function was not a one-to-one function.
Therefore, the inverse of y = x^2 - 4 is y = ±√(x + 4).
As you correctly mentioned, the presence of the square root (√) indicates that the inverse includes even roots. Since the inverse of this function includes both the positive and negative square root, it fails the horizontal line test. Consequently, the inverse, y = ±√(x + 4), is not a function.