The graph shows an exponential function $f(x)$, and it passes through the point $(0, \sqrt{2})$. Based on this, we can derive the general form of the equation of the function.

General form of an exponential function:

The general form of an exponential function is:

$f(x) = ab^x$

Where:

• $a$ is the initial value (the $y$-intercept, when $x = 0$),
• $b$ is the base of the exponential function.

Step 1: Use the given point $(0, \sqrt{2})$

From the graph, we know the function passes through $(0, \sqrt{2})$. This tells us that when $x = 0$, $f(x) = \sqrt{2}$. Plugging this into the general equation:

$f(0) = a \cdot b^0 = a \cdot 1 = a$ So, $a = \sqrt{2}$.

Step 2: Determine $b$

Next, we need to determine $b$. From the graph, it appears that the function is increasing exponentially, meaning $b > 1$. However, we don't have another exact point to directly calculate $b$, but we can hypothesize that the base could be $b = e$, which is a common base for exponential functions.

So, the equation of the exponential function is:

$f(x) = \sqrt{2} \cdot e^x$

This is a reasonable assumption based on the graph's shape and the common use of $e$ in exponential growth problems.

Let me know if you'd like further details or clarifications!

Related questions:

1. How can we verify if $b = e$ is the correct base for this function?
2. What would the equation look like if the base $b$ were not $e$?
3. How does changing the base $b$ affect the steepness of the graph?
4. What is the domain and range of this exponential function?
5. How would the graph change if we multiplied $e^x$ by a constant other than $\sqrt{2}$?

Tip:

When analyzing exponential functions, look for points where $x = 0$ to quickly determine the initial value, and use other points to find the base.