A student writes, "The inverse of y = -√(x+2) is y = x^2 - 2." Why is this statement false?

I don't understand how it isnt the inverse. The textbook says that changing the domain will make it the inverse? Wouldn't that only make it a function?

You need f(f^-1(x)) = x

-√(x^2-2 + 2) = -√(x^2) = -x

this fails because x^2 is not one-to-one

The statement is false because the proposed formula, y = x^2 - 2, does not represent the inverse of the given function, y = -√(x + 2).

To determine if a formula represents the inverse of a function, you need to check if the composition of the two functions equals the identity function. In this case, we would substitute y = -√(x + 2) into y = x^2 - 2 and simplify:

y = x^2 - 2
-√(x + 2) = x^2 - 2
√(x + 2) = -x^2 + 2

By squaring both sides of the equation, we get:

x + 2 = x^4 - 4x^2 + 4

Rearranging the terms, we have:

x^4 - 4x^2 + x + 2 = 0

This equation is not equivalent to the original equation y = -√(x + 2), indicating that y = x^2 - 2 is not the inverse of the given function.

Additionally, changing the domain alone does not make a function the inverse. The inverse of a function not only changes the domain but also interchanges the role of x and y. In other words, if (a, b) is a point on the original function, then (b, a) should be a point on the inverse function.

Therefore, it is important to verify the composition of the functions and the interchange of variables to determine the true inverse of a given function.

The statement is false because the given equation, y = -√(x+2), does not have an inverse function in its current form. Let's examine why:

To find the inverse of a function, we need to switch the roles of x and y, and then solve for the new y (which will be the inverse function). So, let's begin by switching x and y:

x = -√(y+2)

Now, our goal is to isolate y. To do that, we'll need to eliminate the square root term. Squaring both sides of the equation would accomplish this:

x^2 = (-√(y+2))^2
x^2 = y + 2

At this point, we can simply rearrange the equation to solve for y:

y = x^2 - 2

Now, we have arrived at the equation y = x^2 - 2, which is a different equation from the one stated in the student's claim. Therefore, the given equation, y = -√(x+2), and this new equation, y = x^2 - 2, are not inverse functions.

Regarding the part about changing the domain, it's important to note that changing the domain may result in a new function, but it does not necessarily guarantee that it will be the inverse. In this case, changing the domain would only alter the range of the original function, not its inverse.

In summary, the statement is false because the given equation does not have an inverse function in the form y = x^2 - 2.