Problem 32 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\sqrt{x^{2}-49}$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \sqrt{x^2 - 49} \) is \((-\infty, -7] \cup [7, \infty)\).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{x^2 - 49} \). This is a square root function, which means the expression inside the square root, \( x^2 - 49 \), must be greater than or equal to 0 for \( f(x) \) to be real. Identifying where this expression is non-negative will help us determine the domain.

Step 2: Set Inside Expression Non-negative

Set the expression inside the square root to be non-negative: \( x^2 - 49 \geq 0 \). We need to solve this inequality to determine the values of \( x \) that satisfy this condition.

Step 3: Solve the Quadratic Inequality

First, factor the expression: \( x^2 - 49 = (x - 7)(x + 7) \). For the product of two factors to be non-negative, both factors should be either positive or negative. Consider the values where each factor is zero: \( x - 7 = 0 \) or \( x + 7 = 0 \). Thus, \( x = 7 \) or \( x = -7 \) are points of sign change.

Step 4: Determine the Sign of Each Interval

The critical points divide the number line into three intervals: \( (-\infty, -7) \), \( [-7, 7] \), and \( (7, \infty) \). Test a point in each interval to determine where the expression \((x-7)(x+7)\) is non-negative:- Choose \( x = -8 \) for \( (-\infty, -7) \): \((-8-7)(-8+7) = 15\), positive.- Choose \( x = 0 \) for \( [-7, 7] \): \((0-7)(0+7) = -49\), negative.- Choose \( x = 8 \) for \( (7, \infty) \): \((8-7)(8+7) = 15\), positive.

Step 5: Identify the Domain

From the sign determination, the expression \((x-7)(x+7)\) is non-negative in the intervals \((-\infty, -7] \cup [7, \infty)\). Therefore, the domain of \( f(x) \) is all real numbers \( x \) such that \( x \leq -7 \) or \( x \geq 7 \).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is set greater than, less than, greater than or equal to, or less than or equal to another expression, often zero. In this exercise, we are dealing with the inequality:

• \( x^2 - 49 \geq 0 \)

The first step in solving quadratic inequalities is to factor the quadratic expression, if possible. In our case, it factors into:

• \( (x-7)(x+7) \)

The next step is identifying the values of \( x \) where each factor equals zero; these values are called the critical points. Here, the critical points are at \( x = -7 \) and \( x = 7 \). These points divide the number line into intervals, and we check each interval to see if the entire expression is positive or non-negative. For the given problem, only the intervals \((-\infty, -7] \) and \([7, \infty)\) satisfy the inequality. Therefore, \( x \) must fall into these intervals for the quadratic expression to be non-negative.

Square Root Functions

Square root functions are one of the fundamental function types. These functions only make sense when the expression under the square root, known as the radicand, is non-negative since we cannot extract the square root of a negative number in real numbers. The function we consider here is:

• \( f(x) = \sqrt{x^2-49} \)

For this to return real and non-complex values, the radicand must be at least zero, which effectively imposes a restriction on the domain of the function. The inequality set up to find this range is:

• \( x^2 - 49 \geq 0 \)

Solving this inequality, we determined that the values of \( x \) are restricted to the intervals \( x \leq -7 \) or \( x \geq 7 \). When considering square root functions, it is crucial to remember:

• The domain depends on the value of the radicand being non-negative.
• This results in constraints on the values \( x \) can take.
• The domains often include certain intervals or points, derived from where the radicand switches signs.

Factoring Expressions

Factoring is the process of breaking down complex expressions into simpler monomials or polynomials that produce the original expression when multiplied together. In our exercise, we had:

• \( x^2 - 49 \)

Using the difference of squares, which is a common factoring technique, we rewrote this as:

• \( (x-7)(x+7) \)

Recognizing this pattern simplifies solving quadratic inequalities since it reveals the roots of the equation (\( x = -7 \) and \( x = 7 \) in this case). Factoring is a crucial tool in algebra because:

• It simplifies expressions and reveals key properties like roots, which are essential in solving equations or inequalities.
• Once factored, it becomes easier to interpret and solve equations by analyzing the sign of products over different intervals of the independent variable.
• Factoring helps in both simplifying expressions for everyday use as well as solving complex algebraic tasks effectively.