Precalculus Examples

$(\frac{3}{5}, \frac{4}{5})$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{4}{5}$ of the point, and set $x$ to the $x$ value $\frac{3}{5}$ of the point.

$\frac{4}{5} = a^{\frac{3}{5}}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{\frac{3}{5}} = \frac{4}{5}$ .

$a^{\frac{3}{5}} = \frac{4}{5}$

Step 2.2

Raise each side of the equation to the power of $\frac{5}{3}$ to eliminate the fractional exponent on the left side.

${(a^{\frac{3}{5}})}^{\frac{5}{3}} = {(\frac{4}{5})}^{\frac{5}{3}}$

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 ${(a^{\frac{3}{5}})}^{\frac{5}{3}}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${(a^{\frac{3}{5}})}^{\frac{5}{3}}$ .

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Step 2.3.1.1.1.1

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

$a^{\frac{3}{5} \cdot \frac{5}{3}} = {(\frac{4}{5})}^{\frac{5}{3}}$

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

Cancel the common factor.

$a^{\frac{3}{5} \cdot \frac{5}{3}} = {(\frac{4}{5})}^{\frac{5}{3}}$

Step 2.3.1.1.1.2.2

Rewrite the expression.

$a^{\frac{1}{5} \cdot 5} = {(\frac{4}{5})}^{\frac{5}{3}}$

Step 2.3.1.1.1.3

Cancel the common factor of $5$ .

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Step 2.3.1.1.1.3.1

Cancel the common factor.

$a^{\frac{1}{5} \cdot 5} = {(\frac{4}{5})}^{\frac{5}{3}}$

Step 2.3.1.1.1.3.2

Rewrite the expression.

$a^{1} = {(\frac{4}{5})}^{\frac{5}{3}}$

Step 2.3.1.1.2

Simplify.

$a = {(\frac{4}{5})}^{\frac{5}{3}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Apply the product rule to $\frac{4}{5}$ .

$a = \frac{4^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 3

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

$f (x) = {(\frac{4^{\frac{5}{3}}}{5^{\frac{5}{3}}})}^{x}$