Finite Math Examples

$f (x) = - 7 x^{2} + 2 x + 8$

Step 1

Write $f (x) = - 7 x^{2} + 2 x + 8$ as an equation.

$y = - 7 x^{2} + 2 x + 8$

Step 2

Interchange the variables.

$x = - 7 y^{2} + 2 y + 8$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 7 y^{2} + 2 y + 8 = x$ .

$- 7 y^{2} + 2 y + 8 = x$

Step 3.2

Subtract $x$ from both sides of the equation.

$- 7 y^{2} + 2 y + 8 - x = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = - 7$ , $b = 2$ , and $c = 8 - x$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 7 \cdot (8 - x))}}{2 \cdot - 7}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 7 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.5.1.2

Multiply $- 4$ by $- 7$ .

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.5.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot 8 + 28 (- x)}}{2 \cdot - 7}$

Step 3.5.1.4

Multiply $28$ by $8$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 + 28 (- x)}}{2 \cdot - 7}$

Step 3.5.1.5

Multiply $- 1$ by $28$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 - 28 x}}{2 \cdot - 7}$

Step 3.5.1.6

Add $4$ and $224$ .

$y = \frac{- 2 \pm \sqrt{228 - 28 x}}{2 \cdot - 7}$

Step 3.5.1.7

Factor $4$ out of $228 - 28 x$ .

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Step 3.5.1.7.1

Factor $4$ out of $228$ .

$y = \frac{- 2 \pm \sqrt{4 (57) - 28 x}}{2 \cdot - 7}$

Step 3.5.1.7.2

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

$y = \frac{- 2 \pm \sqrt{4 (57) + 4 (- 7 x)}}{2 \cdot - 7}$

Step 3.5.1.7.3

Factor $4$ out of $4 (57) + 4 (- 7 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (57 - 7 x)}}{2 \cdot - 7}$

Step 3.5.1.8

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

$y = \frac{- 2 \pm \sqrt{2^{2} (57 - 7 x)}}{2 \cdot - 7}$

Step 3.5.1.9

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{2 \cdot - 7}$

Step 3.5.2

Multiply $2$ by $- 7$ .

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$

Step 3.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$ .

$y = \frac{1 \pm \sqrt{57 - 7 x}}{7}$

Step 3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 7 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.6.1.2

Multiply $- 4$ by $- 7$ .

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot 8 + 28 (- x)}}{2 \cdot - 7}$

Step 3.6.1.4

Multiply $28$ by $8$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 + 28 (- x)}}{2 \cdot - 7}$

Step 3.6.1.5

Multiply $- 1$ by $28$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 - 28 x}}{2 \cdot - 7}$

Step 3.6.1.6

Add $4$ and $224$ .

$y = \frac{- 2 \pm \sqrt{228 - 28 x}}{2 \cdot - 7}$

Step 3.6.1.7

Factor $4$ out of $228 - 28 x$ .

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Step 3.6.1.7.1

Factor $4$ out of $228$ .

$y = \frac{- 2 \pm \sqrt{4 (57) - 28 x}}{2 \cdot - 7}$

Step 3.6.1.7.2

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

$y = \frac{- 2 \pm \sqrt{4 (57) + 4 (- 7 x)}}{2 \cdot - 7}$

Step 3.6.1.7.3

Factor $4$ out of $4 (57) + 4 (- 7 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (57 - 7 x)}}{2 \cdot - 7}$

Step 3.6.1.8

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

$y = \frac{- 2 \pm \sqrt{2^{2} (57 - 7 x)}}{2 \cdot - 7}$

Step 3.6.1.9

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{2 \cdot - 7}$

Step 3.6.2

Multiply $2$ by $- 7$ .

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$

Step 3.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$ .

$y = \frac{1 \pm \sqrt{57 - 7 x}}{7}$

Step 3.6.4

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{57 - 7 x}}{7}$

Step 3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 7 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.7.1.2

Multiply $- 4$ by $- 7$ .

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot (8 - x)}}{2 \cdot - 7}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 28 \cdot 8 + 28 (- x)}}{2 \cdot - 7}$

Step 3.7.1.4

Multiply $28$ by $8$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 + 28 (- x)}}{2 \cdot - 7}$

Step 3.7.1.5

Multiply $- 1$ by $28$ .

$y = \frac{- 2 \pm \sqrt{4 + 224 - 28 x}}{2 \cdot - 7}$

Step 3.7.1.6

Add $4$ and $224$ .

$y = \frac{- 2 \pm \sqrt{228 - 28 x}}{2 \cdot - 7}$

Step 3.7.1.7

Factor $4$ out of $228 - 28 x$ .

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Step 3.7.1.7.1

Factor $4$ out of $228$ .

$y = \frac{- 2 \pm \sqrt{4 (57) - 28 x}}{2 \cdot - 7}$

Step 3.7.1.7.2

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

$y = \frac{- 2 \pm \sqrt{4 (57) + 4 (- 7 x)}}{2 \cdot - 7}$

Step 3.7.1.7.3

Factor $4$ out of $4 (57) + 4 (- 7 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (57 - 7 x)}}{2 \cdot - 7}$

Step 3.7.1.8

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

$y = \frac{- 2 \pm \sqrt{2^{2} (57 - 7 x)}}{2 \cdot - 7}$

Step 3.7.1.9

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{2 \cdot - 7}$

Step 3.7.2

Multiply $2$ by $- 7$ .

$y = \frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$

Step 3.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{57 - 7 x}}{- 14}$ .

$y = \frac{1 \pm \sqrt{57 - 7 x}}{7}$

Step 3.7.4

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{57 - 7 x}}{7}$

Step 3.8

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{57 - 7 x}}{7}$

$y = \frac{1 - \sqrt{57 - 7 x}}{7}$

$y = \frac{1 + \sqrt{57 - 7 x}}{7}$

$y = \frac{1 - \sqrt{57 - 7 x}}{7}$

Step 4

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

$f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$

Step 5

Verify if $f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$ is the inverse of $f (x) = - 7 x^{2} + 2 x + 8$ .

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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 $f (x) = - 7 x^{2} + 2 x + 8$ and $f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$ and compare them.

Step 5.2

Find the range of $f (x) = - 7 x^{2} + 2 x + 8$ .

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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:

$(- \infty, \frac{57}{7}]$

Step 5.3

Find the domain of $\frac{1 + \sqrt{57 - 7 x}}{7}$ .

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Step 5.3.1

Set the radicand in $\sqrt{57 - 7 x}$ greater than or equal to $0$ to find where the expression is defined.

$57 - 7 x \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Subtract $57$ from both sides of the inequality.

$- 7 x \geq - 57$

Step 5.3.2.2

Divide each term in $- 7 x \geq - 57$ by $- 7$ and simplify.

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Step 5.3.2.2.1

Divide each term in $- 7 x \geq - 57$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 57}{- 7}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 57}{- 7}$

Step 5.3.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 57}{- 7}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{57}{7}$

Step 5.3.3

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

$(- \infty, \frac{57}{7}]$

Step 5.4

Find the domain of $f (x) = - 7 x^{2} + 2 x + 8$ .

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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 $f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$ is the range of $f (x) = - 7 x^{2} + 2 x + 8$ and the range of $f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$ is the domain of $f (x) = - 7 x^{2} + 2 x + 8$ , then $f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$ is the inverse of $f (x) = - 7 x^{2} + 2 x + 8$ .

$f^{- 1} (x) = \frac{1 + \sqrt{57 - 7 x}}{7}, \frac{1 - \sqrt{57 - 7 x}}{7}$

Step 6