Precalculus Examples

$(\frac{1}{2}, \frac{\sqrt{3}}{2})$

Step 1

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{\sqrt{3}}{2}$ of the point, and set $x$ to the $x$ value $\frac{1}{2}$ of the point.

$\frac{\sqrt{3}}{2} = a^{\frac{1}{2}}$

Step 2

Solve the equation for $a$ .

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Step 2.1

Rewrite the equation as $a^{\frac{1}{2}} = \frac{\sqrt{3}}{2}$ .

$a^{\frac{1}{2}} = \frac{\sqrt{3}}{2}$

Step 2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(a^{\frac{1}{2}})}^{2} = {(\frac{\sqrt{3}}{2})}^{2}$

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 ${(a^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${(a^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1.1.1.1

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

$a^{\frac{1}{2} \cdot 2} = {(\frac{\sqrt{3}}{2})}^{2}$

Step 2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {(\frac{\sqrt{3}}{2})}^{2}$

Step 2.3.1.1.1.2.2

Rewrite the expression.

$a^{1} = {(\frac{\sqrt{3}}{2})}^{2}$

Step 2.3.1.1.2

Simplify.

$a = {(\frac{\sqrt{3}}{2})}^{2}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify ${(\frac{\sqrt{3}}{2})}^{2}$ .

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Step 2.3.2.1.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$a = \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 2.3.2.1.2

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

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Step 2.3.2.1.2.1

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

$a = \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 2.3.2.1.2.2

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

$a = \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 2.3.2.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.2.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.2.4.1

Cancel the common factor.

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

Step 2.3.2.1.2.4.2

Rewrite the expression.

$a = \frac{3^{1}}{2^{2}}$

Step 2.3.2.1.2.5

Evaluate the exponent.

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

Step 2.3.2.1.3

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

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

Step 3

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

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