This problem aims to find the **exponential function** of a given curve, and there lies a point on that curve at which the solution will proceed. To better understand the problem, you need to have good knowledge of exponential functions and their **decay **and **growth rate techniques**.

First, let’s discuss what an exponential function is. An **exponential function** is a mathematical function denoted by the expression:

\[ f(x) = exp | e^ x \]

This expression refers to a **positive value function**, or it can also be extended to be **complex numbers**.

But let’s see how we can understand the concept and figure out if an expression is exponential. If there is an increase of 1 in the exponential value of x, the multiplying factor will always be constant. Also, a similar ratio will be observed when you switch from one term to another.

## Expert Answer:

To start with, we are given a point that lies on the curve as shown in the graph figure.

The given point in $x, y$ coordinate system is $(-2, 9)$.

Using our **exponential formula**:

\[ f(x) = a^ x \]

Here, $a$ refers to the exponent with exponential growth factor $x$.

Now simply plug in the value of $x$ from the given point into our mentioned equation. This will give the value of our unknown parameter $. f$.

\[ 9 = a^ {-2} \]

To equalize the left and right-hand sides, we are going to rewrite $9$ so that the exponents become equal, i.e., $3^ 2$, and this gives us:

\[ 3^2 = a^{-2} \]

Further simplifying:

\[ \left( \dfrac{1}{3} \right) ^{-2}= a^{-2} \]

From the above equation, the variable $a$ can be found as $ \left( \dfrac{1}{3} \right) $

Thus, our exponential function turns out to be:

\[ f = \left( \dfrac{1}{3} \right) ^{x} \]

## Numerical Answer

\[ f = \left( \dfrac{1}{3} \right) ^ {x} \]

## Example

Determine the exponential function $g(x) = a^x$ whose graph is given.

The given point in $x,y$ co-ordinate system is $(-4, 16)$

Step $1$ is using our exponential formula:

\[ g(x) = a ^ x \]

Now plug in the value of $x$ from the given point into our formula equation. This will give the value of our unknown parameter $. g$.

\[ 16 = a ^ {-4} \]

We are going to rewrite $16$ so that the exponents become equal i.e. $2^4$, this gives us:

\[ 2 ^ 4 = a ^ {-4} \]

Simplifying:

\[ \left( \dfrac{1}{2} \right) ^ {-4}= a ^ {-4} \]

The variable $a$ can be found as $ \left( \dfrac{1}{2} \right) $.

**Final Answer**

\[ g = \left( \dfrac{1}{2} \right) ^ {x} \]

A few things to note here are that the** exponential function** is important when looking at growth and decay or can be used to determine the **growth rate, decay rate, the time passed, **and **something at the given time.**

*Images/mathematical drawings are created with GeoGebra.*