Problem 16 - Precalculus with Calculus Previews

Question

Find an exponential function \(f(x)\) \(=b^{x}\) such that the graph of \(f\) passes through the given point. $$ (2, e) $$

Step-by-Step Solution

Answer

The exponential function is \( f(x) = e^{x/2} \).

Step 1: Understand the Problem

We need to find an exponential function in the form \( f(x) = b^x \) that passes through the point \((2, e)\). This means that when \( x = 2 \), \( f(2) = e \).

Step 2: Substitute the Point into the Function

Since the expression for the function is \( f(x) = b^x \) and we know it passes through \((2, e)\), we substitute these values into the equation: \( b^2 = e \).

Step 3: Solve for the Base \(b\)

We have the equation \( b^2 = e \). To find \( b \), take the square root of both sides: \( b = \sqrt{e} \).

Step 4: Write the Exponential Function

Now that we have found \( b = \sqrt{e} \), the exponential function is \( f(x) = (\sqrt{e})^x \). Alternatively, it can also be written as \( f(x) = e^{x/2} \).

Key Concepts

Graph of a Function

A graph of a function is like a visual map that shows us how a function behaves. In this context, we are dealing with an exponential function, which is expressed as \( f(x) = b^x \). An exponential graph has a characteristic "swooshing" curve, where the function grows very quickly. For our exercise, we're finding an exponential function which passes through a specific point: \((2, e)\). This means that when \( x = 2 \), the output of our function is \( e \), Euler's number, a special constant approximately equal to 2.718.
Understanding how a point fits on a graph is essential:

• Each point \((x, y)\) on a graph represents the input \(x\) and the corresponding output \(y = f(x)\).
• When we're given a specific point like \((2, e)\), it means that our curve must pass through this point, making it a crucial part of finding our function.
• The graph of any exponential function \( f(x) = b^x \) will show different growth rates based on the base \( b \). For bases greater than 1, the function increases rapidly as \( x \) increases.

Graphing these functions can be insightful in predicting behaviors such as exponential growth, seen in populations and financial contexts.

Solving Equations

Solving equations is all about finding which values satisfy an equation, meaning we want to make the equation true. In our problem, after substituting the known point \((2, e)\) into our function \( f(x) = b^x \), we end up with the equation \( b^2 = e \). This tells us that we need to find a base \( b \) that, when squared, equals \( e \).
To solve \( b^2 = e \), we:

• Take the square root of both sides of the equation.
• This gives us \( b = \sqrt{e} \).
• Because the function passes through \((2, e)\), using known rules of algebra helps us isolate \( b \).

Finding out what satisfies the equation is key in solving mathematical problems:

• This step-by-step methodology allows us to systematically find what the base \( b \) should be to accurately represent our function.
• Understanding how to manipulate and solve equations builds a strong foundation for more complex mathematical tasks.

Base of Exponentials

The base of exponentials, represented as \( b \) in \( f(x) = b^x \), is a very important component of exponential functions. It dictates how quickly the function's output grows as \( x \) increases. In our specific problem, we discovered that our base must be \( b = \sqrt{e} \) in order for the function to pass through the point \((2, e)\).
Let's explore why understanding the base is crucial:

• The base determines the rate of increase or decrease for \( f(x) = b^x \).
• When \( b > 1 \), the function exhibits exponential growth.
• If \( 0 < b < 1 \), the function will show exponential decay instead.

Specifically, for this problem:

• We found \( b = \sqrt{e} \), which leads us to alternative forms like \( f(x) = e^{x/2} \).
• This transformation shows how changing the base can still represent the same behavior of the function as it simplifies the expression.

Getting a strong grip on how the base influences exponential functions empowers us to model real-world phenomena, from natural growth patterns to financial predictions.