Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\log _{4}\left(x^{2}-4 x-21\right)$$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is \( (-\infty, -3) \cup (7, \infty) \).

Step 1: Identify the Logarithmic Function

We are given the function \( f(x) = \log_{4}(x^2 - 4x - 21) \). To find the domain of this logarithmic function, we need to determine where the expression inside the logarithm is positive, as the logarithm of zero or a negative number is undefined.

Step 2: Set the Argument Greater Than Zero

Create the inequality with the inner expression of the logarithm: \( x^2 - 4x - 21 > 0 \). Solving this will tell us where the expression is positive.

Step 3: Solve the Quadratic Inequality

First, find the roots by solving \( x^2 - 4x - 21 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -4 \), and \( c = -21 \). Calculating gives the roots \( x_1 = 7 \) and \( x_2 = -3 \).

Step 4: Determine Intervals of Inequality

The roots divide the number line into intervals: \( (-\infty, -3) \), \( (-3, 7) \), and \( (7, \infty) \). Test a point in each interval to determine if the inequality \( x^2 - 4x - 21 > 0 \) holds true.

Step 5: Test Each Interval

- For \((-\infty, -3)\), choose \(x = -4\). Plugging into the inequality: \((-4)^2 - 4(-4) - 21 = 37\). Since 37 is positive, \((-\infty, -3)\) is in the domain.- For \((-3, 7)\), choose \(x = 0\). \(0^2 - 4(0) - 21 = -21\). Since -21 is not positive, \((-3, 7)\) is not in the domain.- For \((7, \infty)\), choose \(x = 8\). \(8^2 - 4(8) - 21 = 11\). Since 11 is positive, \((7, \infty)\) is in the domain.

Step 6: Determine and Write the Domain

Combining the intervals where the inequality holds true, the domain of \( f(x) \) is \( (-\infty, -3) \cup (7, \infty) \).

Step 7: Verify Graphically (Optional)

Graphing \( y = x^2 - 4x - 21 \) will show it above the x-axis in the intervals \( (-\infty, -3) \) and \( (7, \infty) \), confirming our solution.

Key Concepts

Domain of a Function

When working with logarithmic functions, understanding the domain is crucial. The domain refers to all possible input values for which the function is defined and produces valid outputs. For a logarithmic function like \( f(x) = \log_4(x^2 - 4x - 21) \), we must ensure the argument of the logarithm, \( x^2 - 4x - 21 \), is positive. Logarithms are undefined for zero or negative numbers, so we look for where \( x^2 - 4x - 21 > 0 \). This will give us the domain of the function.

In essence, the domain of any logarithmic function starts with solving the inequality set by its argument (xpression inside the log symbol). In our case, this inequality is a quadratic inequality.

Quadratic Inequality

A quadratic inequality involves expressions where the highest power of variable \( x \) is two. Consider the inequality \( x^2 - 4x - 21 > 0 \). To solve, first find the values of \( x \) that make the expression equal to zero. This helps to identify points that divide the number line into intervals.

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is used to find the roots. For our quadratic \( x^2 - 4x - 21 \), we have:

• \( a = 1, b = -4, c = -21 \)
• The roots are \( x_1 = 7 \) and \( x_2 = -3 \)

These roots divide the number line into intervals: \( (-\infty, -3) \), \( (-3, 7) \), and \( (7, \infty) \). By testing points within each interval, we determine if the inequality is satisfied. The intervals that make the inequality true form the domain of the original logarithmic function.

Graphical Representation

Visualizing quadratic expressions on a graph can provide insights into why certain intervals are part of the domain. Graphing the equation \( y = x^2 - 4x - 21 \) shows how it behaves across different intervals.

• The graph is a parabola opening upwards, due to the positive leading coefficient \( a = 1 \).
• At the roots \( x = -3 \) and \( x = 7 \), the parabola intersects the x-axis.

Observing the graph reveals that above the x-axis, the output is positive. These sections, \( (-\infty, -3) \) and \( (7, \infty) \), confirm the analytical solution of the domain. By checking the graph, you ensure these calculations are accurate, providing a visual confirmation that complements the algebraic approach.