Problem 34 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\sqrt{x^{2}+4 x-32}$$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, -8] \cup [4, \infty)\).

Step 1: Analyze the Function Type

The function provided is a square root function. The expression inside the square root, which is a quadratic expression, must be non-negative for the square root to be defined. Therefore, we need to find when the quadratic expression \(x^2 + 4x - 32\) is greater than or equal to zero.

Step 2: Determine Zeros of the Quadratic Expression

To find when \(x^2 + 4x - 32 \geq 0 \), first find the roots of the equation \(x^2 + 4x - 32 = 0\) by factoring or using the quadratic formula. Factoring, we have:\[x^2 + 4x - 32 = (x + 8)(x - 4) = 0\]This gives us the roots \(x = -8\) and \(x = 4\).

Step 3: Determine the Inequality Intervals

Now that we have the roots, we test intervals determined by these roots to see where \(x^2 + 4x - 32 \geq 0\). The intervals are \((-\infty, -8)\), \([-8, 4]\), and \((4, \infty)\).

Step 4: Test the Intervals

Select test points from each interval to determine where the inequality holds:- Choose \(x = -10\) for \((-\infty, -8)\): \((-10+8)(-10-4) > 0\), so the expression is positive.- Choose \(x = 0\) for \([-8, 4]\): \((0+8)(0-4) = -32 < 0\), so the expression is negative.- Choose \(x = 5\) for \((4, \infty)\): \((5+8)(5-4) > 0\), so the expression is positive.

Step 5: Establish the Domain Based on Test Results

The quadratic expression \(x^2 + 4x - 32\) is non-negative on the intervals \((-\infty, -8] \) and \([4, \infty)\). These intervals represent the domain of the function \( f(x) = \sqrt{x^2 + 4x - 32} \).

Key Concepts

Quadratic Expression

A quadratic expression is a mathematical phrase involving a variable raised to the second power, also known as a second-degree polynomial. In general, quadratic expressions follow the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The example examined here, \(x^2 + 4x - 32\), is a typical quadratic expression.
Quadratic expressions are essential because they help us understand the behavior of parabolas – curves that are symmetric around a central line. These expressions have two possible roots, which can be found by factoring, completing the square, or applying the quadratic formula.
In the exercise, we need the roots of the quadratic expression to determine where it is non-negative. Starting from the equation \((x + 8)(x - 4) = 0\), the roots are \(x = -8\) and \(x = 4\). These roots will help assess the function's domain.

Square Root Function

A square root function involves taking the square root of an expression, often represented as \(f(x) = \sqrt{expression}\). In this exercise, our function is \(f(x) = \sqrt{x^2 + 4x - 32}\). The key property of square root functions is that the expression inside the root must be non-negative, i.e., greater than or equal to zero.
This requirement exists because the square root of a negative number is not defined within the scope of real numbers. Thus, finding the correct domain controls when the square root can "exist" without error.
In our case, the expression \(x^2 + 4x - 32\) must be at least zero. So, our job is to find when this condition is met, ensuring the function remains defined and real.

Inequality Intervals

Analyzing inequality intervals involves determining where a function or expression holds a particular inequality across different sections of its domain. For the expression \(x^2 + 4x - 32\), we look at where it is non-negative. We divide the number line based on its roots: \(x = -8\) and \(x = 4\).
The next step is to choose a test point from each created interval to see if the inequality \(x^2 + 4x - 32 \geq 0\) is satisfied.

• For the interval \((-\infty, -8)\), try \(x = -10\). The result is positive.
• For \([-8, 4]\), test \(x = 0\), yielding a negative result.
• For \((4, \infty)\), test \(x = 5\), finding a positive result.

This testing reveals that the quadratic expression is non-negative in the intervals \((-\infty, -8] \) and \([4, \infty)\). Thus, these intervals compose the domain of the function, where the square root is valid and real.