Problem 96 - A Graphical Approach to Precalculus with Limits

Question

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=4^{x-2}-2$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{5}{2}\). The inequality \(f(x)<0\) holds for \(x<\frac{5}{2}\), and \(f(x)\geq0\) for \(x\geq\frac{5}{2}\).

Step 1: Set the Function to Zero

The first step is to solve the equation \(f(x) = 0\). Given \(f(x) = 4^{x-2} - 2\), set it to zero: \[ 4^{x-2} - 2 = 0 \]

Step 2: Isolate the Exponential Term

Add 2 to both sides of the equation to isolate the exponential term: \[ 4^{x-2} = 2 \]

Step 3: Convert to Logarithmic Form

To solve for \(x\), convert the equation to a logarithmic form. Recognize that \(4^{x-2} = 2\) can be written as \((2^2)^{x-2} = 2^1\), or \(4^{x-2} = 2\). Taking logarithm base 2 of both sides yields, \[ (x-2) \log_2{4} = 1 \]

Step 4: Solve for x

Since \(log_2{4} = 2\), simplify the equation: \((x-2) \cdot 2 = 1\). Solve for \(x\): \[ x-2 = \frac{1}{2} \] \[ x = 2 + \frac{1}{2} \] \[ x = \frac{5}{2} \]

Step 5: Analyze f(x) Graphical Behavior

Now substitute this back to find where \(f(x)<0\) and \(f(x)\geq 0\). Draw the graph of \(f(x) = 4^{x-2} - 2\). Recognize that the function decreases for \(x < \frac{5}{2}\) and increases for \(x > \frac{5}{2}\).

Step 6: Find Inequality Solutions

From the graph, observe:- \(f(x) < 0\) for \(x < \frac{5}{2}\).- \(f(x) \geq 0\) for \(x \geq \frac{5}{2}\).

Key Concepts

Exponential equations

Exponential equations are equations where the variable appears in the exponent. They often take the form of something like \( a^{x} = b \). In our example, we had the equation \( 4^{x-2} - 2 = 0 \). To solve these types of equations, the main steps involve ensuring that both sides of the equation can potentially have the same base.For instance:

• Isolate the exponential term: Get the exponential component by itself on one side of the equation. Here, adding 2 to both sides immediately isolates the term, leading to \( 4^{x-2} = 2 \).
• Change the base if necessary: Since \( 4 = 2^2 \), we can express the equation as \((2^2)^{x-2} = 2^1\).
• Apply logarithms: If both sides still aren't in a base of 2, convert the equation using a logarithm that matches the base. In this case, \((x-2)\log_2{4} = 1\) can be simplified to solve for \(x\).

Exponentials are powerful and simplify many real-world phenomena like compounding interest, population growth, and radioactive decay. Recognizing and solving exponential equations is an essential skill in precalculus.

Graphical analysis

Graphical analysis involves studying a graph to understand the behavior of a function. It is a visual approach to gain insights into where functions increase or decrease, and where they cross the axes. In the exercise we worked on, analyzing the graph played a crucial role in understanding where the function \(f(x) = 4^{x-2} - 2\) exceeded, or stayed below zero.The steps to conduct a graphical analysis are as follows:

• Draw the function: Plotting \(f(x) = 4^{x-2} - 2\) offers a curve starting below the x-axis and increasing as it moves to the right.
• Look for intercepts: The x-intercept was found at \(x = \frac{5}{2}\), derived from solving \(f(x) = 0\). This is where our graphical curve crosses the x-axis.
• Examine behavior: The function decreases to the left of the intercept and increases to the right. This aligns with the negative and positive nature of the function's value relative to the x-axis.

Using a graph helps students visually correlate the results from analytical calculations with their geometric representations, offering a more comprehensive understanding of the function's behavior.

Inequality solutions

Inequality solutions involve finding the range of values for which a function satisfies certain conditions, such as being less than or greater than a particular value. For our function \(f(x) = 4^{x-2} - 2\), we were tasked with resolving \(f(x)<0\) and \(f(x) \geq 0\).The process for solving inequalities involves:

• Define the inequality: Start by determining where the function is either less than zero or greater than or equal to zero. The solution for \(f(x)=0\), i.e., \(x = \frac{5}{2}\), serves as the boundary point.
• Use the graph: From the graphical analysis, observe the function's behavior relative to this boundary. Specifically, \(f(x) < 0\) when the graph is below the x-axis, or for \(x < \frac{5}{2}\).
• Determine intervals: Inequalities like \(f(x) \geq 0\) are true for \(x \geq \frac{5}{2}\), where the graph lies on or above the x-axis.

Solving inequalities is crucial as it allows us to specify exact intervals where certain conditions are met. In real-life applications, these might represent times when a stock remains profitable, the duration temperatures stay stable, or periods of favorable growth.