Algebra Examples

$f (x) = x^{2} - 3$

Step 1

Write $f (x) = x^{2} - 3$ as an equation.

$y = x^{2} - 3$

Step 2

Interchange the variables.

$x = y^{2} - 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y^{2} - 3 = x$ .

$y^{2} - 3 = x$

Step 3.2

Add $3$ to both sides of the equation.

$y^{2} = x + 3$

Step 3.3

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

$y = \pm \sqrt{x + 3}$

Step 3.4

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

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Step 3.4.1

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

$y = \sqrt{x + 3}$

Step 3.4.2

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

$y = - \sqrt{x + 3}$

Step 3.4.3

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

$y = \sqrt{x + 3}$

$y = - \sqrt{x + 3}$

$y = \sqrt{x + 3}$

$y = - \sqrt{x + 3}$

$y = \sqrt{x + 3}$

$y = - \sqrt{x + 3}$

Step 4

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

$f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$

Step 5

Verify if $f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$ is the inverse of $f (x) = x^{2} - 3$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x^{2} - 3$ and $f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$ and compare them.

Step 5.2

Find the range of $f (x) = x^{2} - 3$ .

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Step 5.2.1

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

Interval Notation:

$[- 3, \infty)$

Step 5.3

Find the domain of $\sqrt{x + 3}$ .

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Step 5.3.1

Set the radicand in $\sqrt{x + 3}$ greater than or equal to $0$ to find where the expression is defined.

$x + 3 \geq 0$

Step 5.3.2

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

Step 5.3.3

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

$[- 3, \infty)$

Step 5.4

Find the domain of $f (x) = x^{2} - 3$ .

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Step 5.4.1

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 5.5

Since the domain of $f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$ is the range of $f (x) = x^{2} - 3$ and the range of $f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$ is the domain of $f (x) = x^{2} - 3$ , then $f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$ is the inverse of $f (x) = x^{2} - 3$ .

$f^{- 1} (x) = \sqrt{x + 3}, - \sqrt{x + 3}$

Step 6