Find the Exponential Function (3,-2)

$(3, - 2)$

Step 1

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

$- 2 = a^{3}$

Step 2

Solve the equation for $a$ .

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

$a^{3} = - 2$

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

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

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

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

Rewrite $- 2$ as $- 1 (2)$ .

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

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

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

Pull terms out from under the radical.

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

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

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

Step 3

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

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

Find the Exponential Function (3,-2)

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