Problem 48 - Precalculus

Question

Find the domain of the function. \(f(x)=\log \left(\frac{x^{2}+9 x+18}{4 x-20}\right)\)

Step-by-Step Solution

Answer

Domain: (-∞, -6) ∪ (5, ∞)

Step 1: Identify the Expression Inside the Logarithm

The function is defined as \( f(x) = \log \left( \frac{x^2 + 9x + 18}{4x - 20} \right) \). The expression inside the logarithm must be positive, i.e., \( \frac{x^2 + 9x + 18}{4x - 20} > 0 \).

Step 2: Set the Numerator to Greater Than Zero

Identify when the numerator is positive, \(x^2 + 9x + 18 > 0\). Factor in the numerator: solve \(x^2 + 9x + 18 = 0\), using the factorization \((x + 3)(x + 6)\). The roots are \(x = -3\) and \(x = -6\). Divide the number domain into intervals based on these roots and test each interval to determine where the numerator is positive.

Step 3: Set the Denominator's Conditions

To ensure the value is defined, the denominator cannot be zero: solve \(4x - 20 eq 0\) which leads to \(x eq 5\). Additionally, determine when the expression \(4x - 20 > 0\) to agree with the overall expression being positive by solving \(x > 5\).

Step 4: Determine Sign of Overall Fraction

Combine the results of the numerator and denominator analysis to determine where the entire fraction \(\frac{x^2 + 9x + 18}{4x - 20} > 0\). Consider intervals: \((-\infty, -6)\), \((-6, -3)\), \((-3, 5)\), and \((5, \infty)\), based on critical points \(-6, -3, 5\).

Step 5: Test Intervals

Choose test points in each interval to check where the fraction is positive: - For \((-\infty, -6)\), try \(x = -7\) ensuring positive.- For \((-6, -3)\), try \(x = -5\) leading to negative (excluded).- For \((-3, 5)\), try \(x = 0\) ensuring negative (excluded).- For \((5, \infty)\), try \(x = 6\) and confirm positive.The positive intervals identified: \((-\infty, -6)\) and \((5, \infty)\).

Step 6: Combine Results

The valid domain for the function is all \(x\) values where the expression inside the logarithm is positive. Based on the test results, the domain is \(x \in (-\infty, -6) \cup (5, \infty)\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical functions that represent the inverse operation of exponentiation. The logarithm of a number is the exponent to which a base, typically 10 (common logarithm) or 'e' (natural logarithm), must be raised to produce that number. For example, \[ ext{if} \ a^b = c, \ ext{then} \ ext{log}_a(c) = b. \]A common property of logarithmic functions is that the expression inside the log, known as the argument, must be positive. This property holds because you cannot take the logarithm of a negative number or zero in real-valued mathematics. This requirement arises since no real number exponent, no matter the base, can yield these values.

• The logarithm function processes multiplication as addition; therefore, \[ ext{log}(xy) = \ ext{log}(x) + \ ext{log}(y). \]
• Similarly, division inside logarithms turns into subtraction: \[ \ ext{log}(\frac{x}{y}) = \ ext{log}(x) - \ ext{log}(y). \]

Inequalities

Inequalities are statements that show the relationship between two expressions that are not equal, using symbols for 'greater than', 'less than', 'greater than or equal to', or 'less than or equal to'. Solving inequalities involves finding the set of all values that satisfy the inequality.

When working with logarithmic functions, the argument of the logarithm must fulfill inequality conditions, such as being greater than zero, to be valid. Consider the inequality \[x > y \]This indicates that x is larger than y. If we reverse it to \[ x < y \, \text{then y is greater than x} \. \]Knowing these comparisons is crucial when analyzing the ranges where functions like the one in the exercise are defined. By leveraging inequalities, we can determine intervals of x that keep the function's argument positive. If the expression takes the form \[\frac{a}{b} > 0, \text{the numerator} \ (a) \ \ ext{and denominator} \ (b) \ ext{must either both be positive or both be negative}.\]

Interval Notation

Interval notation is a mathematical notation used to represent a set of numbers enclosed between a lower and upper bound. An interval includes all numbers between these two boundaries. Here are some fundamentals of interval notation:

• Open Interval: \ ((a, b)) represents all numbers between a and b, but not including a and b.
• Closed Interval: \[ [a, b] \]includes all numbers between a and b, including a and b.
• Infinity: Since infinity is a concept rather than a number, intervals using infinity, such as \[(-\infty, 5) \or \ (5, +\infty), \, \ ext{are always open on the side of infinity.} \]

For the problem at hand, the domain result was \((-\infty, -6) \cup (5, \infty)\), \text{which combines} \ \two open intervals. This means taking all x values where x is less than -6 or greater than 5.

Quadratic Equations

Quadratic equations are polynomial equations of the form \[ ax^2 + bx + c = 0, \text{where} \ a, b, ext{and} c \text{are constants and a} \text{is not zero}. \]These equations can often be solved using the factorization method, completing the square, or the quadratic formula, which is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]For the function in the exercise, the numerator of the expression inside the logarithm is a quadratic, \[x^2 + 9x + 18. \]Factorization simplifies this to \((x + 3)(x + 6)\), which allows us to quickly identify the roots of the equation (x = -3, x = -6). Analyzing these roots through a sign analysis enables us to determine when the expression is positive, negative, or zero. Understanding these key points is essential while determining the valid intervals for the variable x.