Finite Math Examples

$s (t) = - 16 t^{2} + 32 t$

Step 1

Write $s (t) = - 16 t^{2} + 32 t$ as an equation.

$y = - 16 t^{2} + 32 t$

Step 2

Interchange the variables.

$t = - 16 y^{2} + 32 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 16 y^{2} + 32 y = t$ .

$- 16 y^{2} + 32 y = t$

Step 3.2

Subtract $t$ from both sides of the equation.

$- 16 y^{2} + 32 y - t = 0$

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 $a = - 16$ , $b = 32$ , and $c = - t$ into the quadratic formula and solve for $y$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (- 16 \cdot (- t))}}{2 \cdot - 16}$

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 $32$ to the power of $2$ .

$y = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.5.1.2

Multiply $- 4 \cdot - 16 \cdot - 1$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $- 16$ .

$y = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.5.1.2.2

Multiply $64$ by $- 1$ .

$y = \frac{- 32 \pm \sqrt{1024 - 64 t}}{2 \cdot - 16}$

Step 3.5.1.3

Factor $64$ out of $1024 - 64 t$ .

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Step 3.5.1.3.1

Factor $64$ out of $1024$ .

$y = \frac{- 32 \pm \sqrt{64 (16) - 64 t}}{2 \cdot - 16}$

Step 3.5.1.3.2

Factor $64$ out of $- 64 t$ .

$y = \frac{- 32 \pm \sqrt{64 (16) + 64 (- t)}}{2 \cdot - 16}$

Step 3.5.1.3.3

Factor $64$ out of $64 (16) + 64 (- t)$ .

$y = \frac{- 32 \pm \sqrt{64 (16 - t)}}{2 \cdot - 16}$

Step 3.5.1.4

Rewrite $64 (16 - t)$ as $8^{2} (4^{2} - t)$ .

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Step 3.5.1.4.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 32 \pm \sqrt{8^{2} (16 - t)}}{2 \cdot - 16}$

Step 3.5.1.4.2

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

$y = \frac{- 32 \pm \sqrt{8^{2} (4^{2} - t)}}{2 \cdot - 16}$

Step 3.5.1.5

Pull terms out from under the radical.

$y = \frac{- 32 \pm 8 \sqrt{4^{2} - t}}{2 \cdot - 16}$

Step 3.5.1.6

Raise $4$ to the power of $2$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{2 \cdot - 16}$

Step 3.5.2

Multiply $2$ by $- 16$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$

Step 3.5.3

Simplify $\frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$ .

$y = \frac{4 \pm \sqrt{16 - t}}{4}$

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 $32$ to the power of $2$ .

$y = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.6.1.2

Multiply $- 4 \cdot - 16 \cdot - 1$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $- 16$ .

$y = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.6.1.2.2

Multiply $64$ by $- 1$ .

$y = \frac{- 32 \pm \sqrt{1024 - 64 t}}{2 \cdot - 16}$

Step 3.6.1.3

Factor $64$ out of $1024 - 64 t$ .

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Step 3.6.1.3.1

Factor $64$ out of $1024$ .

$y = \frac{- 32 \pm \sqrt{64 (16) - 64 t}}{2 \cdot - 16}$

Step 3.6.1.3.2

Factor $64$ out of $- 64 t$ .

$y = \frac{- 32 \pm \sqrt{64 (16) + 64 (- t)}}{2 \cdot - 16}$

Step 3.6.1.3.3

Factor $64$ out of $64 (16) + 64 (- t)$ .

$y = \frac{- 32 \pm \sqrt{64 (16 - t)}}{2 \cdot - 16}$

Step 3.6.1.4

Rewrite $64 (16 - t)$ as $8^{2} (4^{2} - t)$ .

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Step 3.6.1.4.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 32 \pm \sqrt{8^{2} (16 - t)}}{2 \cdot - 16}$

Step 3.6.1.4.2

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

$y = \frac{- 32 \pm \sqrt{8^{2} (4^{2} - t)}}{2 \cdot - 16}$

Step 3.6.1.5

Pull terms out from under the radical.

$y = \frac{- 32 \pm 8 \sqrt{4^{2} - t}}{2 \cdot - 16}$

Step 3.6.1.6

Raise $4$ to the power of $2$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{2 \cdot - 16}$

Step 3.6.2

Multiply $2$ by $- 16$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$

Step 3.6.3

Simplify $\frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$ .

$y = \frac{4 \pm \sqrt{16 - t}}{4}$

Step 3.6.4

Change the $\pm$ to $+$ .

$y = \frac{4 + \sqrt{16 - t}}{4}$

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 $32$ to the power of $2$ .

$y = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.7.1.2

Multiply $- 4 \cdot - 16 \cdot - 1$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $- 16$ .

$y = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (- 1 t)}}{2 \cdot - 16}$

Step 3.7.1.2.2

Multiply $64$ by $- 1$ .

$y = \frac{- 32 \pm \sqrt{1024 - 64 t}}{2 \cdot - 16}$

Step 3.7.1.3

Factor $64$ out of $1024 - 64 t$ .

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Step 3.7.1.3.1

Factor $64$ out of $1024$ .

$y = \frac{- 32 \pm \sqrt{64 (16) - 64 t}}{2 \cdot - 16}$

Step 3.7.1.3.2

Factor $64$ out of $- 64 t$ .

$y = \frac{- 32 \pm \sqrt{64 (16) + 64 (- t)}}{2 \cdot - 16}$

Step 3.7.1.3.3

Factor $64$ out of $64 (16) + 64 (- t)$ .

$y = \frac{- 32 \pm \sqrt{64 (16 - t)}}{2 \cdot - 16}$

Step 3.7.1.4

Rewrite $64 (16 - t)$ as $8^{2} (4^{2} - t)$ .

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Step 3.7.1.4.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 32 \pm \sqrt{8^{2} (16 - t)}}{2 \cdot - 16}$

Step 3.7.1.4.2

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

$y = \frac{- 32 \pm \sqrt{8^{2} (4^{2} - t)}}{2 \cdot - 16}$

Step 3.7.1.5

Pull terms out from under the radical.

$y = \frac{- 32 \pm 8 \sqrt{4^{2} - t}}{2 \cdot - 16}$

Step 3.7.1.6

Raise $4$ to the power of $2$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{2 \cdot - 16}$

Step 3.7.2

Multiply $2$ by $- 16$ .

$y = \frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$

Step 3.7.3

Simplify $\frac{- 32 \pm 8 \sqrt{16 - t}}{- 32}$ .

$y = \frac{4 \pm \sqrt{16 - t}}{4}$

Step 3.7.4

Change the $\pm$ to $-$ .

$y = \frac{4 - \sqrt{16 - t}}{4}$

Step 3.8

The final answer is the combination of both solutions.

$y = \frac{4 + \sqrt{16 - t}}{4}$

$y = \frac{4 - \sqrt{16 - t}}{4}$

$y = \frac{4 + \sqrt{16 - t}}{4}$

$y = \frac{4 - \sqrt{16 - t}}{4}$

Step 4

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

$s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$

Step 5

Verify if $s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$ is the inverse of $s (t) = - 16 t^{2} + 32 t$ .

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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 $s (t) = - 16 t^{2} + 32 t$ and $s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$ and compare them.

Step 5.2

Find the range of $s (t) = - 16 t^{2} + 32 t$ .

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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:

$(- \infty, 16]$

Step 5.3

Find the domain of $\frac{4 + \sqrt{16 - t}}{4}$ .

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Step 5.3.1

Set the radicand in $\sqrt{16 - t}$ greater than or equal to $0$ to find where the expression is defined.

$16 - t \geq 0$

Step 5.3.2

Solve for $t$ .

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Step 5.3.2.1

Subtract $16$ from both sides of the inequality.

$- t \geq - 16$

Step 5.3.2.2

Divide each term in $- t \geq - 16$ by $- 1$ and simplify.

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Step 5.3.2.2.1

Divide each term in $- t \geq - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- t}{- 1} \leq \frac{- 16}{- 1}$

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.

$\frac{t}{1} \leq \frac{- 16}{- 1}$

Step 5.3.2.2.2.2

Divide $t$ by $1$ .

$t \leq \frac{- 16}{- 1}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Divide $- 16$ by $- 1$ .

$t \leq 16$

Step 5.3.3

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

$(- \infty, 16]$

Step 5.4

Find the domain of $s (t) = - 16 t^{2} + 32 t$ .

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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 $s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$ is the range of $s (t) = - 16 t^{2} + 32 t$ and the range of $s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$ is the domain of $s (t) = - 16 t^{2} + 32 t$ , then $s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$ is the inverse of $s (t) = - 16 t^{2} + 32 t$ .

$s^{- 1} (t) = \frac{4 + \sqrt{16 - t}}{4}, \frac{4 - \sqrt{16 - t}}{4}$

Step 6