Problem 17 - Precalculus with Calculus Previews

Question

Find an exponential function \(f(x)\) \(=b^{x}\) such that the graph of \(f\) passes through the given point. $$ (-2,9) $$

Step-by-Step Solution

Answer

The function is \(f(x) = \left(\frac{1}{3}\right)^x\).

Step 1: Understand the Exponential Function Form

An exponential function is given by the formula \( f(x) = b^x \), where \( b \) is the base of the exponent. Our task is to find the base \( b \) such that the function passes through a specific point.

Step 2: Set Up the Equation with the Given Point

We substitute the coordinates of the given point \((-2, 9)\) into the formula \(f(x) = b^x\). This means replacing \(x\) with \(-2\) and \(f(x)\) with \(9\), giving us the equation \(b^{-2} = 9\).

Step 3: Solve for the Base b

The equation \(b^{-2} = 9\) can be rewritten as \(\frac{1}{b^2} = 9\). Taking reciprocals of both sides, we have \(b^2 = \frac{1}{9}\). Solving for \(b\), we find \(b = \pm \frac{1}{3}\).

Step 4: Determine the Valid Base for an Exponential Function

Exponential functions typically have positive bases. Therefore, we choose \(b = \frac{1}{3}\) as the valid base. Thus, the function \(f(x) = \left(\frac{1}{3}\right)^x\) satisfies the conditions.

Key Concepts

Understanding Graph Analysis in Exponential Functions

When analyzing the graph of an exponential function like \( f(x) = b^x \), we focus on the behaviors of the graph as it passes through given points. Graph analysis in this context means understanding how changes in the base \( b \) affect the shape and direction of the graph.
Exponential functions are known for their characteristic shapes:

• A base \( b \) greater than 1 results in a rapidly increasing curve as \( x \) increases. This is called exponential growth.
• A base \( 0 < b < 1 \) results in a rapidly decreasing curve as \( x \) increases. This is referred to as exponential decay.

For the function \( f(x) = \left(\frac{1}{3}\right)^x \), because \( b = \frac{1}{3} \), the graph will showcase exponential decay. The graph will pass through the point \((-2, 9)\), and it will get closer to the x-axis as \( x \) increases but never touch it. This is an important aspect of graph analysis: identifying how the function behaves and intersects given points.

Steps in Solving Exponential Equations

Solving exponential equations involves finding the unknown variable that makes the equation true. In this exercise, we were tasked with finding the base \( b \) of the exponential function \( f(x) = b^x \) that satisfies the condition given by the point \((-2, 9)\).
Here's a simple breakdown of the process:

• Firstly, insert the point's coordinates into the function: you replace \( x \) with \(-2\) and \( f(x) \) with \( 9 \).
• This gives you the equation \( b^{-2} = 9 \). Rewrite this as \( \frac{1}{b^2} = 9 \).
• To solve for \( b \), take the reciprocal of both sides: \( b^2 = \frac{1}{9} \).
• The last step is finding \( b \) by taking the square root: \( b = \pm \frac{1}{3} \).

Since exponential functions need a positive base, we choose \( b = \frac{1}{3} \). This step-by-step approach demonstrates the methodical way in which we solve exponential equations effectively.

Exploring Function Properties of Exponential Functions

Exponential functions such as \( f(x) = b^x \) have distinctive properties that make them unique from other types of functions. Understanding these properties can aid in both graph analysis and solving equations.
Some key properties include:

• **Domain and Range**: The domain includes all real numbers, but the range is positive real numbers for \( b > 1 \) because exponential functions do not produce zero or negative values.
• **Intercepts**: These functions typically have a y-intercept at \( (0, 1) \) since any number raised to the power of zero equals one, provided \( b > 0 \).
• **Asymptotic Behavior**: Exponential functions approach but never actually reach the x-axis. This line is called a horizontal asymptote at \( y = 0 \).
• **End Behavior**: If \( 0 < b < 1 \), the function decreases towards zero as \( x \) increases, indicating decay. For \( b > 1 \), it increases infinitely as \( x \) increases, indicating growth.

By understanding these properties, we can predict how an exponential function will behave without graphing every point, making us more efficient in solving real-world problems.