Find the Exponential Function (2,1/16)

$(2, \frac{1}{16})$

To find an exponential function, $f (x) = a^{x}$ , containing the point, set $f (x)$ in the function to the $y$ value $\frac{1}{16}$ of the point, and set $x$ to the $x$ value $2$ of the point.

$\frac{1}{16} = a^{2}$

Solve the equation for $a$ .

Rewrite the equation as $a^{2} = \frac{1}{16}$ .

$a^{2} = \frac{1}{16}$

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

$a = \pm \sqrt{\frac{1}{16}}$

Simplify $\pm \sqrt{\frac{1}{16}}$ .

Rewrite $\sqrt{\frac{1}{16}}$ as $\frac{\sqrt{1}}{\sqrt{16}}$ .

$a = \pm \frac{\sqrt{1}}{\sqrt{16}}$

Any root of $1$ is $1$ .

$a = \pm \frac{1}{\sqrt{16}}$

Simplify the denominator.

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

$a = \pm \frac{1}{\sqrt{4^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$a = \pm \frac{1}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$a = \frac{1}{4}$

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \frac{1}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

$a = \frac{1}{4}, - \frac{1}{4}$

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

$f (x) = {(\frac{1}{4})}^{x}, f (x) = {(- \frac{1}{4})}^{x}$

Find the Exponential Function (2,1/16)

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