Finite Math Examples

$f (x) = 6 - 5 x^{2}$

Step 1

Write $f (x) = 6 - 5 x^{2}$ as an equation.

$y = 6 - 5 x^{2}$

Step 2

Interchange the variables.

$x = 6 - 5 y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 - 5 y^{2} = x$ .

$6 - 5 y^{2} = x$

Step 3.2

Subtract $6$ from both sides of the equation.

$- 5 y^{2} = x - 6$

Step 3.3

Divide each term in $- 5 y^{2} = x - 6$ by $- 5$ and simplify.

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Step 3.3.1

Divide each term in $- 5 y^{2} = x - 6$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{x}{- 5} + \frac{- 6}{- 5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{x}{- 5} + \frac{- 6}{- 5}$

Step 3.3.2.1.2

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

$y^{2} = \frac{x}{- 5} + \frac{- 6}{- 5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{x}{5} + \frac{- 6}{- 5}$

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - \frac{x}{5} + \frac{6}{5}$

Step 3.4

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

$y = \pm \sqrt{- \frac{x}{5} + \frac{6}{5}}$

Step 3.5

Simplify $\pm \sqrt{- \frac{x}{5} + \frac{6}{5}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- x + 6}{5}}$

Step 3.5.2

Rewrite $\sqrt{\frac{- x + 6}{5}}$ as $\frac{\sqrt{- x + 6}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- x + 6}}{\sqrt{5}}$

Step 3.5.3

Multiply $\frac{\sqrt{- x + 6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- x + 6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt{- x + 6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.5.4.2

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.5.4.3

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.5.4.4

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.5.4.6

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

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Step 3.5.4.6.1

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.5.4.6.2

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.5.4.6.3

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

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.5.4.6.4

Cancel the common factor of $2$ .

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Step 3.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{5^{1}}$

Step 3.5.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{- x + 6} \sqrt{5}}{5}$

Step 3.5.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(- x + 6) \cdot 5}}{5}$

Step 3.5.6

Reorder factors in $\pm \frac{\sqrt{(- x + 6) \cdot 5}}{5}$ .

$y = \pm \frac{\sqrt{5 (- x + 6)}}{5}$

Step 3.6

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

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Step 3.6.1

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

$y = \frac{\sqrt{5 (- x + 6)}}{5}$

Step 3.6.2

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

$y = - \frac{\sqrt{5 (- x + 6)}}{5}$

Step 3.6.3

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

$y = \frac{\sqrt{5 (- x + 6)}}{5}$

$y = - \frac{\sqrt{5 (- x + 6)}}{5}$

$y = \frac{\sqrt{5 (- x + 6)}}{5}$

$y = - \frac{\sqrt{5 (- x + 6)}}{5}$

$y = \frac{\sqrt{5 (- x + 6)}}{5}$

$y = - \frac{\sqrt{5 (- x + 6)}}{5}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$ is the inverse of $f (x) = 6 - 5 x^{2}$ .

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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 $f (x) = 6 - 5 x^{2}$ and $f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$ and compare them.

Step 5.2

Find the range of $f (x) = 6 - 5 x^{2}$ .

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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:

$(- \infty, 6]$

Step 5.3

Find the domain of $\frac{\sqrt{5 (- x + 6)}}{5}$ .

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Step 5.3.1

Set the radicand in $\sqrt{5 (- x + 6)}$ greater than or equal to $0$ to find where the expression is defined.

$5 (- x + 6) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $5 (- x + 6) \geq 0$ by $5$ and simplify.

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Step 5.3.2.1.1

Divide each term in $5 (- x + 6) \geq 0$ by $5$ .

$\frac{5 (- x + 6)}{5} \geq \frac{0}{5}$

Step 5.3.2.1.2

Simplify the left side.

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Step 5.3.2.1.2.1

Cancel the common factor of $5$ .

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Step 5.3.2.1.2.1.1

Cancel the common factor.

$\frac{5 (- x + 6)}{5} \geq \frac{0}{5}$

Step 5.3.2.1.2.1.2

Divide $- x + 6$ by $1$ .

$- x + 6 \geq \frac{0}{5}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $5$ .

$- x + 6 \geq 0$

Step 5.3.2.2

Subtract $6$ from both sides of the inequality.

$- x \geq - 6$

Step 5.3.2.3

Divide each term in $- x \geq - 6$ by $- 1$ and simplify.

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Step 5.3.2.3.1

Divide each term in $- x \geq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 6}{- 1}$

Step 5.3.2.3.2

Simplify the left side.

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Step 5.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 6}{- 1}$

Step 5.3.2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 1}$

Step 5.3.2.3.3

Simplify the right side.

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Step 5.3.2.3.3.1

Divide $- 6$ by $- 1$ .

$x \leq 6$

Step 5.3.3

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

$(- \infty, 6]$

Step 5.4

Find the domain of $f (x) = 6 - 5 x^{2}$ .

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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 $f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$ is the range of $f (x) = 6 - 5 x^{2}$ and the range of $f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$ is the domain of $f (x) = 6 - 5 x^{2}$ , then $f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$ is the inverse of $f (x) = 6 - 5 x^{2}$ .

$f^{- 1} (x) = \frac{\sqrt{5 (- x + 6)}}{5}, - \frac{\sqrt{5 (- x + 6)}}{5}$

Step 6