Math Snap

STEP 1

What is this asking?
We're given a function $f(x)$ that takes an $x$ and gives us a $y$, and we need to find the inverse function, written $f^{-1}(y)$, which does the opposite: it takes a $y$ and gives us back the original $x$.
Watch out!
Remember, the inverse function undoes what the original function does!
Don't mix up $x$ and $y$ when finding the inverse.
Also, pay close attention to any restrictions on the domain of $y$.

STEP 2

1. Swap x and y
2. Solve for y
3. Verify the inverse

STEP 3

We're starting with $y = \sqrt{x - 1}$.
This tells us how $y$ depends on $x$.

STEP 4

To find the inverse, we swap $x$ and $y$!
This gives us $x = \sqrt{y - 1}$.
Now we have $x$ in terms of $y$, which is what we want for the inverse function.

STEP 5

To get $y$ by itself, we need to square both sides of the equation $x = \sqrt{y - 1}$.
Squaring a square root gets rid of the root, so we have $x^2 = y - 1$.
Remember, we're doing this to isolate $y$.

STEP 6

Now, we add one to both sides of the equation to isolate $y$.
This gives us $x^2 + 1 = y$.
Almost there!

STEP 7

Notice that in the original equation, $y = \sqrt{x - 1}$, the square root is only defined for non-negative values.
This means $y$ must be greater than or equal to zero.
Since we swapped $x$ and $y$, this restriction now applies to $x$ in our inverse function.
So, we have $x \ge 0$.

STEP 8

We found that $y = x^2 + 1$ with $x \ge 0$.
Since the inverse function is a function of $y$, we rewrite this as $f^{-1}(y) = y^2 + 1$ with $y \ge 0$.

STEP 9

This matches answer choice (B)!

The inverse function is $f^{-1}(y) = y^2 + 1$, where $y \ge 0$.
So the answer is (B).