Finite Math Examples

$(3, - \frac{5}{2})$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $- \frac{5}{2}$ of the point, and set $x$ to the $x$ value $3$ of the point.

$- \frac{5}{2} = a^{3}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{3} = - \frac{5}{2}$ .

$a^{3} = - \frac{5}{2}$

Step 2.2

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

$a = \sqrt[3]{- \frac{5}{2}}$

Step 2.3

Simplify $\sqrt[3]{- \frac{5}{2}}$ .

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Step 2.3.1

Rewrite $- \frac{5}{2}$ as ${({(- 1)}^{3})}^{3} \frac{5}{2}$ .

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Step 2.3.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$a = \sqrt[3]{{(- 1)}^{3} \frac{5}{2}}$

Step 2.3.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$a = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{5}{2}}$

Step 2.3.2

Pull terms out from under the radical.

$a = {(- 1)}^{3} \sqrt[3]{\frac{5}{2}}$

Step 2.3.3

Raise $- 1$ to the power of $3$ .

$a = - \sqrt[3]{\frac{5}{2}}$

Step 2.3.4

Rewrite $\sqrt[3]{\frac{5}{2}}$ as $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ .

$a = - \frac{\sqrt[3]{5}}{\sqrt[3]{2}}$

Step 2.3.5

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$a = - (\frac{\sqrt[3]{5}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 2.3.6

Combine and simplify the denominator.

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Step 2.3.6.1

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 2.3.6.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 2.3.6.3

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

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 2.3.6.4

Add $1$ and $2$ .

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 2.3.6.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 2.3.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 2.3.6.5.2

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

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 2.3.6.5.3

Combine $\frac{1}{3}$ and $3$ .

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.3.6.5.4

Cancel the common factor of $3$ .

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Step 2.3.6.5.4.1

Cancel the common factor.

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 2.3.6.5.4.2

Rewrite the expression.

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 2.3.6.5.5

Evaluate the exponent.

$a = - \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{2}$

Step 2.3.7

Simplify the numerator.

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Step 2.3.7.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$a = - \frac{\sqrt[3]{5} \sqrt[3]{2^{2}}}{2}$

Step 2.3.7.2

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

$a = - \frac{\sqrt[3]{5} \sqrt[3]{4}}{2}$

Step 2.3.8

Simplify the numerator.

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Step 2.3.8.1

Combine using the product rule for radicals.

$a = - \frac{\sqrt[3]{5 \cdot 4}}{2}$

Step 2.3.8.2

Multiply $5$ by $4$ .

$a = - \frac{\sqrt[3]{20}}{2}$

Step 3

Substitute each value for $a$ back into the function $f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(- \frac{\sqrt[3]{20}}{2})}^{x}$