Problem 72 - College Algebra

Question

Find the domain of each function. $$ f(x)=\sqrt{x^{2}-5 x-24} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \sqrt{x^{2}-5x-24}\) is \(-3 \leq x < 8\) and \(8 \leq x < \infty\).

Step 1: Write down the inequality

Since the values inside the square root must be greater than or equal to 0, an inequality can be written as follows: \(x^{2}-5x-24 \geq 0\)

Step 2: Factorize the quadratic expression

The next step involves factorizing the quadratic expression so it can be solved for x. This will give two numbers p and q such that \(p*q = -24\) and \(p + q = -5\). In this case, factorizing the expression gives: \((x-8)(x+3) \geq 0\).

Step 3: Solve the inequality

To solve the inequality \((x-8)(x+3) \geq 0\), it's important to consider where the expression is positive, negative or zero. This happens when \(x-8 = 0\) or \(x+3 = 0\). Solving for x gives two possible values: \(x = 8\) and \(x = -3\). So, the intervals where the inequality could be positive are: \(-\infty \leq x < -3\), \(-3 \leq x < 8\) and \(8 \leq x < \infty\). By analyzing the signs, only \(-3 \leq x < 8\) and \(8 \leq x < \infty\) are true.

Key Concepts

Inequality Solving

When you're tasked with solving inequalities, your goal is to determine where a given expression holds true. In this case, the expression is inside a square root, which means it must be non-negative. To achieve this, first write an inequality:

• The basic requirement is for the expression under the square root to be greater than or equal to zero: \( x^2 - 5x - 24 \geq 0 \).
• This ensures that the square root value is real.

Solving inequalities involves finding intervals for the variable \( x \) where this condition holds. By analyzing these intervals, you can figure out where the entire expression is valid. Breaking down the problem into stages like identifying critical points and testing intervals is the key step.

Quadratic Factorization

Factorizing quadratics is crucial in simplifying expressions, making it easier to solve inequalities. In the original problem, the quadratic expression is given by:

• You need to factorize: \( x^2 - 5x - 24 \).
• Factorization involves writing the expression as a product of two binomials: \( (x - 8)(x + 3) \).
• This is derived by finding values of \( p \) and \( q \) such that \( p \times q = -24 \) (the constant term) and \( p + q = -5 \) (the coefficient of \( x \)).

By factorizing the quadratic expression, you can clearly see its roots, \( x = 8 \) and \( x = -3 \), which are crucial to solving the inequality.

Square Root Function

A square root function is fundamentally about determining the non-negative part of a value. Therefore, it can only work with non-negative inputs. For functions of the form \( f(x) = \sqrt{something} \):

• The 'something' must be greater than or equal to zero, producing real-number solutions only for non-negative inputs.
• The range of the square root function, like \( f(x) = \sqrt{x^2 - 5x - 24} \), is dependent on the domain of the expression inside the root.

Understanding these constraints can help in correctly identifying the domain where the function is defined and ensures no complex numbers are involved in solution sets.

Interval Notation

Interval notation is a concise method to express a range of values. It is necessary to denote the domain of functions. In the process of solving the inequality:

• Use bracket symbols; square brackets \([ ]\) mean the endpoint is included, while parentheses \(( )\) mean it is not.
• For the expression \( \sqrt{x^2 - 5x - 24} \geq 0 \), the intervals were determined as \([-3, 8) \cup [8, \infty)\).
• This notation clearly indicates the regions where the function is valid.

Such precise representation simplifies comprehending function domains and is vital in describing solutions efficiently.