Problem 100 - A Graphical Approach to College Algebra

Question

For the given \(f(x)\), solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\) $$f(x)=9 \log _{3}(3 x)-18$$

Step-by-Step Solution

Answer

Solution: \(x = 3\); \(f(x)<0\) for \(x<3\), \(f(x)\geq 0\) for \(x\geq 3\).

Step 1: Simplify the Equation

The given function is \[f(x) = 9 \log_{3}(3x) - 18\]To solve \(f(x) = 0\), start by adding \(18\) to both sides of the equation:\[9 \log_{3}(3x) = 18\]

Step 2: Divide by 9

Next, divide both sides of the equation by \(9\) to isolate the logarithmic term:\[\log_{3}(3x) = 2\]

Step 3: Convert the Logarithmic Equation

Convert the logarithmic equation to its exponential form. Recall that \(\log_{b}(a) = c\) means \(b^{c} = a\). Thus, we have:\[3^{2} = 3x\]

Step 4: Solve for x

Solve the resulting equation for \(x\):\[9 = 3x\]Divide both sides by \(3\):\[x = 3\]

Step 5: Analyze the Graph for Inequalities

To solve the inequalities \(f(x) < 0\) and \(f(x) \geq 0\), use the graph of \(y = f(x)\). Since \(f(x) = 9 \log_{3}(3x) - 18\) equals zero at \(x = 3\), determine where the function is positive or negative by analyzing the behavior of the logarithm around \(x = 3\). The graph of \(y = 9 \log_{3}(3x)\) is increasing, so:- For \(x > 3\), \(\log_{3}(3x) > 2\), so \(f(x) > 0\) - For \(x < 3\), \(\log_{3}(3x) < 2\), so \(f(x) < 0\)Thus, \(f(x) < 0\) for \(x < 3\) and \(f(x) \geq 0\) for \(x \geq 3\).

Step 6: Conclusion

The solution to the equation \(f(x) = 0\) is \(x = 3\). The inequality \(f(x) < 0\) holds for \(x < 3\), and \(f(x) \geq 0\) holds for \(x \geq 3\).

Key Concepts

Solving Equations

To solve the equation step-by-step, we start with the function \( f(x) = 9 \log_{3}(3x) - 18 \). Our goal is to find when this function equals zero, or in mathematical terms, find \( x \) such that \( f(x) = 0 \). Begin by isolating the logarithmic term on one side of the equation. You add 18 to both sides to get:

• \( 9 \log_{3}(3x) = 18 \)

To simplify further, divide each side by 9:

• \( \log_{3}(3x) = 2 \)

At this point, it's useful to remember that logarithms can be converted to their exponential form. The equation \( \log_{3}(3x) = 2 \) can be rewritten as \( 3^2 = 3x \). This step is crucial because it translates the logarithmic equation into a more familiar form:

• \( 9 = 3x \)

Finally, solve for \( x \) by dividing both sides by 3, which gives:

• \( x = 3 \)

We've discovered that \( x = 3 \) is the solution to the equation, meaning the logarithmic function equals zero at this point.

Inequalities

With inequalities involving a function such as \( f(x) = 9 \log_{3}(3x) - 18 \), there may be regions where the function is less than or greater than zero. Solving these involves understanding the behavior of the function around the values found. For the inequality \( f(x) < 0 \), we need to determine when the function graphs below the x-axis.
First note the critical point from solving the equation: \( x = 3 \). Since we used \( x = 3 \) in solving the equality, it's where the function crosses or touches the x-axis. Around this point, determine where the function resides relative to the x-axis:

• For \( x < 3 \): \( \log_{3}(3x) < 2 \) leads to \( f(x) < 0 \).
• For \( x > 3 \): \( \log_{3}(3x) > 2 \) leading to \( f(x) > 0 \).

This tells us that for \( x < 3 \), the function is negative. Consequently, \( f(x) < 0 \) holds true in this region. Conversely, for \( f(x) \geq 0 \), solutions exist for \( x \geq 3 \) including \( x = 3 \) where the function equals zero.

Graph Analysis

Examining the function graphically helps in confirming our analytical solutions and provides visual insight into the behavior of \( f(x) = 9 \log_{3}(3x) - 18 \). Graph analysis involves plotting the function's output, examining its shape, intercepts, and identifying increasing or decreasing trends.
The function \( f(x) \) is composed of a logarithmic part \( 9 \log_{3}(3x) \), subtracted by 18. Logarithmic functions typically increase slowly and are defined for positive real numbers. At \( x = 3 \), our solution, the function meets the x-axis which means the graph passes through the point (3,0). To its left, as \( x \to 0^+ \), the logarithm approaches \(-\infty\), thus making the function negative.
Observing the graph at \( x > 3 \), see that \( \log_{3}(3x) \) increases with increasing \( x \). As a result, \( f(x) \) makes a corresponding increase, turning positive. This confirms the calculated inequality solutions and shows where \( f(x) \) maintains positiveness or negativeness based on its increasing nature after crossing the x-axis at the point \( x = 3 \). The visual representation reaffirms:

• Negative for \( x < 3 \)
• Positive for \( x \geq 3 \)