Problem 95 - 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 \(f(x) \geq 0\). $$f(x)=2\left(3^{x}\right)-18$$

Step-by-Step Solution

Answer

The equation solves to \(x = 2\). \(f(x) < 0\) for \(x < 2\) and \(f(x) \geq 0\) for \(x \geq 2\).

Step 1: Set up the Equation to Solve for x

The given function is \(f(x) = 2 \cdot 3^{x} - 18\). We need to find when this function equals zero, which means solving the equation \(2 \cdot 3^{x} - 18 = 0\).

Step 2: Isolate the Exponential Term

Move 18 to the other side of the equation: \(2 \cdot 3^{x} = 18\). Then, divide both sides by 2 to isolate the exponential term: \(3^{x} = 9\).

Step 3: Solve the Exponential Equation

Recognize that 9 can be expressed as a power of 3: \(3^2 = 9\). Therefore, \(3^{x} = 3^{2}\). By comparing the exponents, \(x = 2\). This is the solution for the equation \(f(x) = 0\).

Step 4: Analyze the Function Graphically

To analyze the inequalities, consider the graph of \(y = f(x) = 2 \cdot 3^{x} - 18\). The graph of \(y = 3^{x}\) is an exponential curve that grows rapidly. Multiplying by 2 and subtracting 18 adjusts the scale and vertical shift, respectively.

Step 5: Determine When f(x) < 0 and f(x) \(\geq\) 0

Since we found that \(f(x) = 0\) when \(x = 2\), and knowing the shape of the exponential curve, \(f(x) < 0\) for \(x < 2\) because the exponential part grows as x increases. Thus, \(f(x) \geq 0\) for \(x \geq 2\).

Key Concepts

Understanding Inequalities with Exponential Equations

When dealing with exponential equations like \( f(x) = 2 \cdot 3^{x} - 18 \), we often need to solve inequalities such as \( f(x) < 0 \) or \( f(x) \geq 0 \). This requires understanding where the graph of the function is located in relation to the x-axis.

To solve \( f(x) < 0 \), you're asking: For what values of \( x \) does the function fall below the x-axis? From the original analysis, we know that \( f(x) = 0 \) when \( x = 2 \). Therefore, \( f(x) < 0 \) for all \( x < 2 \).

• Identify points where \( f(x) = 0 \) — these points are critical as they separate the graph into regions.
• Exponential curves such as \( 3^x \) grow fast; they will cross the x-axis only once if there is a single solution.
• Regions of the graph indicate intervals where \( f(x) \) is negative or positive.

Using Graphical Analysis to Solve Equations

Graphical analysis is a powerful tool for understanding the behavior of a function, especially exponential ones. When analyzing \( f(x) = 2 \cdot 3^{x} - 18 \), start by noticing that the function is a transformed version of the base exponential function \( y = 3^x \).

• Vertical Stretch: Multiplying by 2 stretches the graph vertically, making it steeper.
• Vertical Shift: Subtracting 18 shifts the entire graph 18 units downwards.

By sketching or viewing this graph, you can see where the function intersects the x-axis at \( x = 2 \). For \( x < 2 \), the graph is below the x-axis, and for \( x \geq 2 \), it is above or touches it.

Graphical representation not only shows solutions but gives insight into why they occur, reinforcing analytical results.

Deep Dive into Function Analysis

Function analysis of \( f(x) = 2 \cdot 3^{x} - 18 \) involves looking at key features of the function, which influence how it behaves across its domain.

Domain and Range

The domain of this exponential function is all real numbers, \( x \in (-\infty, \infty) \), because \( 3^x \) is defined for any real number. The range is \((-18, \infty)\) since \( 2 \cdot 3^x \) can grow indefinitely, but it is always more than \(-18\).

Intercepts and Critial Points

The critical point where the function changes behavior is \( x = 2 \). Here, \( f(x) = 0 \), and this intercept is crucial for establishing the solution intervals for the inequalities.

Analyzing Behavior

With exponential growth, after the intercept, the function value \( f(x) \) becomes increasingly positive as \( 3^x \) grows. Below the critical value \( x < 2 \), the function struggles to increase to zero, thus remaining negative.