Find the Inverse g(x)=-(x^2)/4+7

$g (x) = - \frac{x^{2}}{4} + 7$

Step 1

Write $g (x) = - \frac{x^{2}}{4} + 7$ as an equation.

$y = - \frac{x^{2}}{4} + 7$

Step 2

Interchange the variables.

$x = - \frac{y^{2}}{4} + 7$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{y^{2}}{4} + 7 = x$ .

$- \frac{y^{2}}{4} + 7 = x$

Subtract $7$ from both sides of the equation.

$- \frac{y^{2}}{4} = x - 7$

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{y^{2}}{4}) = - 4 (x - 7)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 4 (- \frac{y^{2}}{4})$ .

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{y^{2}}{4}$ into the numerator.

$- 4 \frac{- y^{2}}{4} = - 4 (x - 7)$

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- y^{2}}{4} = - 4 (x - 7)$

Cancel the common factor.

$4 \cdot - 1 \frac{- y^{2}}{4} = - 4 (x - 7)$

Rewrite the expression.

$- - y^{2} = - 4 (x - 7)$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y^{2} = - 4 (x - 7)$

Multiply $y^{2}$ by $1$ .

$y^{2} = - 4 (x - 7)$

Simplify the right side.

Simplify $- 4 (x - 7)$ .

Apply the distributive property.

$y^{2} = - 4 x - 4 \cdot - 7$

Multiply $- 4$ by $- 7$ .

$y^{2} = - 4 x + 28$

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

$y = \pm \sqrt{- 4 x + 28}$

Simplify $\pm \sqrt{- 4 x + 28}$ .

Factor $4$ out of $- 4 x + 28$ .

Factor $4$ out of $- 4 x$ .

$y = \pm \sqrt{4 (- x) + 28}$

Factor $4$ out of $28$ .

$y = \pm \sqrt{4 (- x) + 4 (7)}$

Factor $4$ out of $4 (- x) + 4 (7)$ .

$y = \pm \sqrt{4 (- x + 7)}$

Rewrite $4$ as $2^{2}$ .

$y = \pm \sqrt{2^{2} (- x + 7)}$

Pull terms out from under the radical.

$y = \pm 2 \sqrt{- x + 7}$

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.

$y = 2 \sqrt{- x + 7}$

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

$y = - 2 \sqrt{- x + 7}$

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

$y = 2 \sqrt{- x + 7}$

$y = - 2 \sqrt{- x + 7}$

$y = 2 \sqrt{- x + 7}$

$y = - 2 \sqrt{- x + 7}$

$y = 2 \sqrt{- x + 7}$

$y = - 2 \sqrt{- x + 7}$

Step 4

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

$g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$

Step 5

Verify if $g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$ is the inverse of $g (x) = - \frac{x^{2}}{4} + 7$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $g (x) = - \frac{x^{2}}{4} + 7$ and $g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$ and compare them.

Find the range of $g (x) = - \frac{x^{2}}{4} + 7$ .

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

Interval Notation:

$(- \infty, 7]$

Find the domain of $2 \sqrt{- x + 7}$ .

Set the radicand in $\sqrt{- x + 7}$ greater than or equal to $0$ to find where the expression is defined.

$- x + 7 \geq 0$

Solve for $x$ .

Subtract $7$ from both sides of the inequality.

$- x \geq - 7$

Divide each term in $- x \geq - 7$ by $- 1$ and simplify.

Divide each term in $- x \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 7}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 1}$

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x \leq 7$

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

$(- \infty, 7]$

Find the domain of $g (x) = - \frac{x^{2}}{4} + 7$ .

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 $g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$ is the range of $g (x) = - \frac{x^{2}}{4} + 7$ and the range of $g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$ is the domain of $g (x) = - \frac{x^{2}}{4} + 7$ , then $g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$ is the inverse of $g (x) = - \frac{x^{2}}{4} + 7$ .

$g^{- 1} (x) = 2 \sqrt{- x + 7}, - 2 \sqrt{- x + 7}$

Step 6

Find the Inverse g(x)=-(x^2)/4+7

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