Finite Math Examples

$(- \frac{1}{3}, \frac{7}{6})$

Step 1

To find an exponential function,

$f (x) = a^{x}$ , containing the point, set

$f (x)$ in the function to the

$y$ value

$\frac{7}{6}$ of the point, and set

$x$ to the

$x$ value

$- \frac{1}{3}$ of the point.

$\frac{7}{6} = a^{- \frac{1}{3}}$

Step 2

Solve the equation for

$a$ .

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Step 2.1

Rewrite the equation as

$a^{- \frac{1}{3}} = \frac{7}{6}$ .

$a^{- \frac{1}{3}} = \frac{7}{6}$

Step 2.2

Raise each side of the equation to the power of

$- 3$ to eliminate the fractional exponent on the left side.

${(a^{- \frac{1}{3}})}^{- 3} = {(\frac{7}{6})}^{- 3}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify

${(a^{- \frac{1}{3}})}^{- 3}$ .

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Step 2.3.1.1.1

Multiply the exponents in

${(a^{- \frac{1}{3}})}^{- 3}$ .

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Step 2.3.1.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$a^{- \frac{1}{3} \cdot - 3} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.1.2

Cancel the common factor of

$3$ .

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Step 2.3.1.1.1.2.1

Move the leading negative in

$- \frac{1}{3}$ into the numerator.

$a^{\frac{- 1}{3} \cdot - 3} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.1.2.2

Factor

$3$ out of

$- 3$ .

$a^{\frac{- 1}{3} \cdot (3 (- 1))} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.1.2.3

Cancel the common factor.

$a^{\frac{- 1}{3} \cdot (3 \cdot - 1)} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.1.2.4

Rewrite the expression.

$a^{- 1 \cdot - 1} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.1.3

Multiply

$- 1$ by

$- 1$ .

$a^{1} = {(\frac{7}{6})}^{- 3}$

Step 2.3.1.1.2

Simplify.

$a = {(\frac{7}{6})}^{- 3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify

${(\frac{7}{6})}^{- 3}$ .

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Step 2.3.2.1.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$a = {(\frac{6}{7})}^{3}$

Step 2.3.2.1.2

Apply the product rule to

$\frac{6}{7}$ .

$a = \frac{6^{3}}{7^{3}}$

Step 2.3.2.1.3

Raise

$6$ to the power of

$3$ .

$a = \frac{216}{7^{3}}$

Step 2.3.2.1.4

Raise

$7$ to the power of

$3$ .

$a = \frac{216}{343}$

Step 3

Substitute each value for

$a$ back into the function

$f (x) = a^{x}$ to find each possible exponential function.

$f (x) = {(\frac{216}{343})}^{x}$