Algebra Examples

$y = {(x + 1)}^{2}$

Step 1

Interchange the variables.

$x = {(y + 1)}^{2}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as ${(y + 1)}^{2} = x$ .

${(y + 1)}^{2} = x$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y + 1 = \pm \sqrt{x}$

Step 2.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y + 1 = \sqrt{x}$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$y = \sqrt{x} - 1$

Step 2.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$y + 1 = - \sqrt{x}$

Step 2.3.4

Subtract $1$ from both sides of the equation.

$y = - \sqrt{x} - 1$

Step 2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{x} - 1$

$y = - \sqrt{x} - 1$

$y = \sqrt{x} - 1$

$y = - \sqrt{x} - 1$

$y = \sqrt{x} - 1$

$y = - \sqrt{x} - 1$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$

Step 4

Verify if $f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$ is the inverse of $f (x) = {(x + 1)}^{2}$ .

Tap for more steps...

Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = {(x + 1)}^{2}$ and $f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$ and compare them.

Step 4.2

Find the range of $f (x) = {(x + 1)}^{2}$ .

Tap for more steps...

Step 4.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Step 4.3

Find the domain of $\sqrt{x} - 1$ .

Tap for more steps...

Step 4.3.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 4.3.2

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 4.4

Find the domain of $f (x) = {(x + 1)}^{2}$ .

Tap for more steps...

Step 4.4.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 4.5

Since the domain of $f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$ is the range of $f (x) = {(x + 1)}^{2}$ and the range of $f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$ is the domain of $f (x) = {(x + 1)}^{2}$ , then $f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$ is the inverse of $f (x) = {(x + 1)}^{2}$ .

$f^{- 1} (x) = \sqrt{x} - 1, - \sqrt{x} - 1$

Step 5