Find the Inverse f(x)=500(0.04-x^2)

$f (x) = 500 (0.04 - x^{2})$

Write $f (x) = 500 (0.04 - x^{2})$ as an equation.

$y = 500 (0.04 - x^{2})$

Interchange the variables.

$x = 500 (0.04 - y^{2})$

Solve for $y$ .

Rewrite the equation as $500 (0.04 - y^{2}) = x$ .

$500 (0.04 - y^{2}) = x$

Divide each term in $500 (0.04 - y^{2}) = x$ by $500$ and simplify.

Divide each term in $500 (0.04 - y^{2}) = x$ by $500$ .

$\frac{500 (0.04 - y^{2})}{500} = \frac{x}{500}$

Simplify the left side.

Cancel the common factor of $500$ .

Cancel the common factor.

$\frac{500 (0.04 - y^{2})}{500} = \frac{x}{500}$

Divide $0.04 - y^{2}$ by $1$ .

$0.04 - y^{2} = \frac{x}{500}$

Subtract $0.04$ from both sides of the equation.

$- y^{2} = \frac{x}{500} - 0.04$

Divide each term in $- y^{2} = \frac{x}{500} - 0.04$ by $- 1$ and simplify.

Divide each term in $- y^{2} = \frac{x}{500} - 0.04$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{\frac{x}{500}}{- 1} + \frac{- 0.04}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{\frac{x}{500}}{- 1} + \frac{- 0.04}{- 1}$

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

$y^{2} = \frac{\frac{x}{500}}{- 1} + \frac{- 0.04}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{\frac{x}{500}}{- 1}$ .

$y^{2} = - 1 \cdot \frac{x}{500} + \frac{- 0.04}{- 1}$

Rewrite $- 1 \cdot \frac{x}{500}$ as $- \frac{x}{500}$ .

$y^{2} = - \frac{x}{500} + \frac{- 0.04}{- 1}$

Divide $- 0.04$ by $- 1$ .

$y^{2} = - \frac{x}{500} + 0.04$

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

$y = \pm \sqrt{- \frac{x}{500} + 0.04}$

Simplify $\pm \sqrt{- \frac{x}{500} + 0.04}$ .

Factor $0.04$ out of $- \frac{x}{500} + 0.04$ .

Factor $0.04$ out of $- \frac{x}{500}$ .

$y = \pm \sqrt{0.04 (- 25 \frac{x}{500}) + 0.04}$

Factor $0.04$ out of $0.04$ .

$y = \pm \sqrt{0.04 (- 25 \frac{x}{500}) + 0.04 (1)}$

Factor $0.04$ out of $0.04 (- 25 \frac{x}{500}) + 0.04 (1)$ .

$y = \pm \sqrt{0.04 (- 25 \frac{x}{500} + 1)}$

Cancel the common factor of $25$ .

Factor $25$ out of $- 25$ .

$y = \pm \sqrt{0.04 (25 (- 1) \frac{x}{500} + 1)}$

Factor $25$ out of $500$ .

$y = \pm \sqrt{0.04 (25 \cdot - 1 \frac{x}{25 \cdot 20} + 1)}$

Cancel the common factor.

$y = \pm \sqrt{0.04 (25 \cdot - 1 \frac{x}{25 \cdot 20} + 1)}$

Rewrite the expression.

$y = \pm \sqrt{0.04 (- 1 \frac{x}{20} + 1)}$

Simplify the expression.

Rewrite $- 1 \frac{x}{20}$ as $- \frac{x}{20}$ .

$y = \pm \sqrt{0.04 (- \frac{x}{20} + 1)}$

Write $1$ as a fraction with a common denominator.

$y = \pm \sqrt{0.04 (- \frac{x}{20} + \frac{20}{20})}$

Combine the numerators over the common denominator.

$y = \pm \sqrt{0.04 \frac{- x + 20}{20}}$

Combine $0.04$ and $\frac{- x + 20}{20}$ .

$y = \pm \sqrt{\frac{0.04 (- x + 20)}{20}}$

Factor $20$ out of $20$ .

$y = \pm \sqrt{\frac{0.04 (- x + 20)}{20 (1)}}$

Separate fractions.

$y = \pm \sqrt{\frac{0.04}{20} \cdot \frac{- x + 20}{1}}$

Divide $0.04$ by $20$ .

$y = \pm \sqrt{0.002 \frac{- x + 20}{1}}$

Cancel the common factor of $0.002$ .

Factor $0.002$ out of $1$ .

$y = \pm \sqrt{0.002 \frac{- x + 20}{0.002 \cdot 500}}$

Cancel the common factor.

$y = \pm \sqrt{0.002 \frac{- x + 20}{0.002 \cdot 500}}$

Rewrite the expression.

$y = \pm \sqrt{\frac{- x + 20}{500}}$

Rewrite $\frac{- x + 20}{500}$ as ${(\frac{1}{10})}^{2} \frac{- x + 20}{5}$ .

Factor the perfect power $1^{2}$ out of $- x + 20$ .

$y = \pm \sqrt{\frac{1^{2} (- x + 20)}{500}}$

Factor the perfect power $10^{2}$ out of $500$ .

$y = \pm \sqrt{\frac{1^{2} (- x + 20)}{10^{2} \cdot 5}}$

Rearrange the fraction $\frac{1^{2} (- x + 20)}{10^{2} \cdot 5}$ .

$y = \pm \sqrt{{(\frac{1}{10})}^{2} \frac{- x + 20}{5}}$

Pull terms out from under the radical.

$y = \pm \frac{1}{10} \sqrt{\frac{- x + 20}{5}}$

Rewrite $\sqrt{\frac{- x + 20}{5}}$ as $\frac{\sqrt{- x + 20}}{\sqrt{5}}$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20}}{\sqrt{5}}$

Multiply $\frac{\sqrt{- x + 20}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{1}{10} (\frac{\sqrt{- x + 20}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{- x + 20}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{- x + 20} \sqrt{5}}{5}$

Combine using the product rule for radicals.

$y = \pm \frac{1}{10} \cdot \frac{\sqrt{(- x + 20) \cdot 5}}{5}$

Multiply $\frac{1}{10} \cdot \frac{\sqrt{(- x + 20) \cdot 5}}{5}$ .

Multiply $\frac{1}{10}$ by $\frac{\sqrt{(- x + 20) \cdot 5}}{5}$ .

$y = \pm \frac{\sqrt{(- x + 20) \cdot 5}}{10 \cdot 5}$

Multiply $10$ by $5$ .

$y = \pm \frac{\sqrt{(- x + 20) \cdot 5}}{50}$

Reorder factors in $\pm \frac{\sqrt{(- x + 20) \cdot 5}}{50}$ .

$y = \pm \frac{\sqrt{5 (- x + 20)}}{50}$

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.

$y = \frac{\sqrt{5 (- x + 20)}}{50}$

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

$y = - \frac{\sqrt{5 (- x + 20)}}{50}$

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

$y = \frac{\sqrt{5 (- x + 20)}}{50}$

$y = - \frac{\sqrt{5 (- x + 20)}}{50}$

$y = \frac{\sqrt{5 (- x + 20)}}{50}$

$y = - \frac{\sqrt{5 (- x + 20)}}{50}$

$y = \frac{\sqrt{5 (- x + 20)}}{50}$

$y = - \frac{\sqrt{5 (- x + 20)}}{50}$

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

$f^{- 1} (x) = \frac{\sqrt{5 (- x + 20)}}{50}, - \frac{\sqrt{5 (- x + 20)}}{50}$

Verify if $f^{- 1} (x) = \frac{\sqrt{5 (- x + 20)}}{50}, - \frac{\sqrt{5 (- x + 20)}}{50}$ is the inverse of $f (x) = 500 (0.04 - x^{2})$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 500 (0.04 - x^{2})$ and $f^{- 1} (x) = \frac{\sqrt{5 (- x + 20)}}{50}, - \frac{\sqrt{5 (- x + 20)}}{50}$ and compare them.

Find the range of $f (x) = 500 (0.04 - x^{2})$ .

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

Interval Notation:

$(- \infty, 20]$

Find the domain of $\frac{\sqrt{5 (- x + 20)}}{50}$ .

Set the radicand in $\sqrt{5 (- x + 20)}$ greater than or equal to $0$ to find where the expression is defined.

$5 (- x + 20) \geq 0$

Solve for $x$ .

Divide each term in $5 (- x + 20) \geq 0$ by $5$ and simplify.

Divide each term in $5 (- x + 20) \geq 0$ by $5$ .

$\frac{5 (- x + 20)}{5} \geq \frac{0}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 (- x + 20)}{5} \geq \frac{0}{5}$

Divide $- x + 20$ by $1$ .

$- x + 20 \geq \frac{0}{5}$

Simplify the right side.

Divide $0$ by $5$ .

$- x + 20 \geq 0$

Subtract $20$ from both sides of the inequality.

$- x \geq - 20$

Divide each term in $- x \geq - 20$ by $- 1$ and simplify.

Divide each term in $- x \geq - 20$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 20}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 20}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 20}{- 1}$

Simplify the right side.

Divide $- 20$ by $- 1$ .

$x \leq 20$

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

$(- \infty, 20]$

Since the domain of $f^{- 1} (x) = \frac{\sqrt{5 (- x + 20)}}{50}, - \frac{\sqrt{5 (- x + 20)}}{50}$ is not equal to the range of $f (x) = 500 (0.04 - x^{2})$ , then $f^{- 1} (x) = \frac{\sqrt{5 (- x + 20)}}{50}, - \frac{\sqrt{5 (- x + 20)}}{50}$ is not an inverse of $f (x) = 500 (0.04 - x^{2})$ .

There is no inverse

Find the Inverse f(x)=500(0.04-x^2)

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