Finite Math Examples

$f (x) = x (- 4 x + 180)$

Step 1

Write $f (x) = x (- 4 x + 180)$ as an equation.

$y = x (- 4 x + 180)$

Step 2

Interchange the variables.

$x = y (- 4 y + 180)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y (- 4 y + 180) = x$ .

$y (- 4 y + 180) = x$

Step 3.2

Simplify $y (- 4 y + 180)$ .

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Step 3.2.1

Simplify by multiplying through.

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Step 3.2.1.1

Apply the distributive property.

$y (- 4 y) + y \cdot 180 = x$

Step 3.2.1.2

Reorder.

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Step 3.2.1.2.1

Rewrite using the commutative property of multiplication.

$- 4 y \cdot y + y \cdot 180 = x$

Step 3.2.1.2.2

Move $180$ to the left of $y$ .

$- 4 y \cdot y + 180 \cdot y = x$

Step 3.2.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.2.1

Move $y$ .

$- 4 (y \cdot y) + 180 \cdot y = x$

Step 3.2.2.2

Multiply $y$ by $y$ .

$- 4 y^{2} + 180 \cdot y = x$

$- 4 y^{2} + 180 y = x$

Step 3.3

Subtract $x$ from both sides of the equation.

$- 4 y^{2} + 180 y - x = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = - 4$ , $b = 180$ , and $c = - x$ into the quadratic formula and solve for $y$ .

$\frac{- 180 \pm \sqrt{180^{2} - 4 \cdot (- 4 \cdot (- x))}}{2 \cdot - 4}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $180$ to the power of $2$ .

$y = \frac{- 180 \pm \sqrt{32400 - 4 \cdot - 4 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.6.1.2

Multiply $- 4 \cdot - 4 \cdot - 1$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $- 4$ .

$y = \frac{- 180 \pm \sqrt{32400 + 16 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.6.1.2.2

Multiply $16$ by $- 1$ .

$y = \frac{- 180 \pm \sqrt{32400 - 16 x}}{2 \cdot - 4}$

Step 3.6.1.3

Factor $16$ out of $32400 - 16 x$ .

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Step 3.6.1.3.1

Factor $16$ out of $32400$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) - 16 x}}{2 \cdot - 4}$

Step 3.6.1.3.2

Factor $16$ out of $- 16 x$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) + 16 (- x)}}{2 \cdot - 4}$

Step 3.6.1.3.3

Factor $16$ out of $16 (2025) + 16 (- x)$ .

$y = \frac{- 180 \pm \sqrt{16 (2025 - x)}}{2 \cdot - 4}$

Step 3.6.1.4

Rewrite $16 (2025 - x)$ as ${(2^{2})}^{2} (2025 - x)$ .

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Step 3.6.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 180 \pm \sqrt{4^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.6.1.4.2

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

$y = \frac{- 180 \pm \sqrt{{(2^{2})}^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.6.1.5

Pull terms out from under the radical.

$y = \frac{- 180 \pm 2^{2} \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.6.1.6

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

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.6.2

Multiply $2$ by $- 4$ .

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$

Step 3.6.3

Simplify $\frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$ .

$y = \frac{45 \pm \sqrt{2025 - x}}{2}$

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 $180$ to the power of $2$ .

$y = \frac{- 180 \pm \sqrt{32400 - 4 \cdot - 4 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.7.1.2

Multiply $- 4 \cdot - 4 \cdot - 1$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $- 4$ .

$y = \frac{- 180 \pm \sqrt{32400 + 16 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.7.1.2.2

Multiply $16$ by $- 1$ .

$y = \frac{- 180 \pm \sqrt{32400 - 16 x}}{2 \cdot - 4}$

Step 3.7.1.3

Factor $16$ out of $32400 - 16 x$ .

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Step 3.7.1.3.1

Factor $16$ out of $32400$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) - 16 x}}{2 \cdot - 4}$

Step 3.7.1.3.2

Factor $16$ out of $- 16 x$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) + 16 (- x)}}{2 \cdot - 4}$

Step 3.7.1.3.3

Factor $16$ out of $16 (2025) + 16 (- x)$ .

$y = \frac{- 180 \pm \sqrt{16 (2025 - x)}}{2 \cdot - 4}$

Step 3.7.1.4

Rewrite $16 (2025 - x)$ as ${(2^{2})}^{2} (2025 - x)$ .

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Step 3.7.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 180 \pm \sqrt{4^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.7.1.4.2

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

$y = \frac{- 180 \pm \sqrt{{(2^{2})}^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.7.1.5

Pull terms out from under the radical.

$y = \frac{- 180 \pm 2^{2} \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.7.1.6

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

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.7.2

Multiply $2$ by $- 4$ .

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$

Step 3.7.3

Simplify $\frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$ .

$y = \frac{45 \pm \sqrt{2025 - x}}{2}$

Step 3.7.4

Change the $\pm$ to $+$ .

$y = \frac{45 + \sqrt{2025 - x}}{2}$

Step 3.8

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

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $180$ to the power of $2$ .

$y = \frac{- 180 \pm \sqrt{32400 - 4 \cdot - 4 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.8.1.2

Multiply $- 4 \cdot - 4 \cdot - 1$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $- 4$ .

$y = \frac{- 180 \pm \sqrt{32400 + 16 \cdot (- 1 x)}}{2 \cdot - 4}$

Step 3.8.1.2.2

Multiply $16$ by $- 1$ .

$y = \frac{- 180 \pm \sqrt{32400 - 16 x}}{2 \cdot - 4}$

Step 3.8.1.3

Factor $16$ out of $32400 - 16 x$ .

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Step 3.8.1.3.1

Factor $16$ out of $32400$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) - 16 x}}{2 \cdot - 4}$

Step 3.8.1.3.2

Factor $16$ out of $- 16 x$ .

$y = \frac{- 180 \pm \sqrt{16 (2025) + 16 (- x)}}{2 \cdot - 4}$

Step 3.8.1.3.3

Factor $16$ out of $16 (2025) + 16 (- x)$ .

$y = \frac{- 180 \pm \sqrt{16 (2025 - x)}}{2 \cdot - 4}$

Step 3.8.1.4

Rewrite $16 (2025 - x)$ as ${(2^{2})}^{2} (2025 - x)$ .

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Step 3.8.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 180 \pm \sqrt{4^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.8.1.4.2

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

$y = \frac{- 180 \pm \sqrt{{(2^{2})}^{2} (2025 - x)}}{2 \cdot - 4}$

Step 3.8.1.5

Pull terms out from under the radical.

$y = \frac{- 180 \pm 2^{2} \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.8.1.6

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

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{2 \cdot - 4}$

Step 3.8.2

Multiply $2$ by $- 4$ .

$y = \frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$

Step 3.8.3

Simplify $\frac{- 180 \pm 4 \sqrt{2025 - x}}{- 8}$ .

$y = \frac{45 \pm \sqrt{2025 - x}}{2}$

Step 3.8.4

Change the $\pm$ to $-$ .

$y = \frac{45 - \sqrt{2025 - x}}{2}$

Step 3.9

The final answer is the combination of both solutions.

$y = \frac{45 + \sqrt{2025 - x}}{2}$

$y = \frac{45 - \sqrt{2025 - x}}{2}$

$y = \frac{45 + \sqrt{2025 - x}}{2}$

$y = \frac{45 - \sqrt{2025 - x}}{2}$

Step 4

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

$f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$ is the inverse of $f (x) = x (- 4 x + 180)$ .

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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 $f (x) = x (- 4 x + 180)$ and $f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$ and compare them.

Step 5.2

Find the range of $f (x) = x (- 4 x + 180)$ .

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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:

$(- \infty, 2025]$

Step 5.3

Find the domain of $\frac{45 + \sqrt{2025 - x}}{2}$ .

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Step 5.3.1

Set the radicand in $\sqrt{2025 - x}$ greater than or equal to $0$ to find where the expression is defined.

$2025 - x \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Subtract $2025$ from both sides of the inequality.

$- x \geq - 2025$

Step 5.3.2.2

Divide each term in $- x \geq - 2025$ by $- 1$ and simplify.

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Step 5.3.2.2.1

Divide each term in $- x \geq - 2025$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 2025}{- 1}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 2025}{- 1}$

Step 5.3.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 2025}{- 1}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Divide $- 2025$ by $- 1$ .

$x \leq 2025$

Step 5.3.3

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

$(- \infty, 2025]$

Step 5.4

Find the domain of $f (x) = x (- 4 x + 180)$ .

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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 $f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$ is the range of $f (x) = x (- 4 x + 180)$ and the range of $f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$ is the domain of $f (x) = x (- 4 x + 180)$ , then $f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$ is the inverse of $f (x) = x (- 4 x + 180)$ .

$f^{- 1} (x) = \frac{45 + \sqrt{2025 - x}}{2}, \frac{45 - \sqrt{2025 - x}}{2}$

Step 6