Find the Inverse f(x)=x^2-4

$f (x) = x^{2} - 4$

Step 1

Write $f (x) = x^{2} - 4$ as an equation.

$y = x^{2} - 4$

Step 2

Interchange the variables.

$x = y^{2} - 4$

Step 3

Solve for $y$ .

Rewrite the equation as $y^{2} - 4 = x$ .

$y^{2} - 4 = x$

Add $4$ to both sides of the equation.

$y^{2} = x + 4$

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

$y = \pm \sqrt{x + 4}$

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

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

$y = \sqrt{x + 4}$

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

$y = - \sqrt{x + 4}$

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

$y = \sqrt{x + 4}$

$y = - \sqrt{x + 4}$

$y = \sqrt{x + 4}$

$y = - \sqrt{x + 4}$

$y = \sqrt{x + 4}$

$y = - \sqrt{x + 4}$

Step 4

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

$f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$

Step 5

Verify if $f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$ is the inverse of $f (x) = x^{2} - 4$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x^{2} - 4$ and $f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$ and compare them.

Find the range of $f (x) = x^{2} - 4$ .

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

Interval Notation:

$[- 4, \infty)$

Find the domain of $\sqrt{x + 4}$ .

Set the radicand in $\sqrt{x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$x + 4 \geq 0$

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

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

$[- 4, \infty)$

Find the domain of $f (x) = x^{2} - 4$ .

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.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$ is the range of $f (x) = x^{2} - 4$ and the range of $f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$ is the domain of $f (x) = x^{2} - 4$ , then $f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$ is the inverse of $f (x) = x^{2} - 4$ .

$f^{- 1} (x) = \sqrt{x + 4}, - \sqrt{x + 4}$

Step 6

Find the Inverse f(x)=x^2-4

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