Math / Algebra

QuestionQuestion 14
The function $f$ is one-to-one. Find the inverse of $f$ as a function of $y$ . $y=f(x)=\sqrt{x-1}$ (A) $f-1(y)=\sqrt{y+1}$ (B) $f-1(y)=y^{2}+1, y \geq 0$ (C) $f-1(y)=(y-1) 2$ (D) $f-1(y)=y^{2}-1, y \geq 0$

Studdy Solution

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)!

STEP 10

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

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QuestionQuestion 14
The function $f$ is one-to-one. Find the inverse of $f$ as a function of $y$ . $y=f(x)=\sqrt{x-1}$ (A) $f-1(y)=\sqrt{y+1}$ (B) $f-1(y)=y^{2}+1, y \geq 0$ (C) $f-1(y)=(y-1) 2$ (D) $f-1(y)=y^{2}-1, y \geq 0$

Studdy Solution

STEP 1

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)!

STEP 10

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

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Studdy Solution

STEP 1

STEP 2

1. Swap x and y2. Solve for y3. Verify the inverse

STEP 3

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

STEP 4

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

STEP 5

To get yyy by itself, we need to **square both sides** of the equation x=y−1x = \sqrt{y - 1}x=y−1​.Squaring a square root gets rid of the root, so we have x2=y−1x^2 = y - 1x2=y−1.Remember, we're doing this to *isolate* yyy.

STEP 6

Now, we **add one to both sides** of the equation to isolate yyy.This gives us x2+1=yx^2 + 1 = yx2+1=y.Almost there!

STEP 7

STEP 8

We found that y=x2+1y = x^2 + 1y=x2+1 with x≥0x \ge 0x≥0.Since the inverse function is a function of yyy, we rewrite this as f−1(y)=y2+1f^{-1}(y) = y^2 + 1f−1(y)=y2+1 with y≥0y \ge 0y≥0.

STEP 9

This matches answer choice (B)!

STEP 10

The inverse function is f−1(y)=y2+1f^{-1}(y) = y^2 + 1f−1(y)=y2+1, where y≥0y \ge 0y≥0.So the answer is (B).

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

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

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.

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 $y$ .

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

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 $y \ge 0$ .

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