Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of \(x\) in the expression. $$\sqrt[4]{3 x^{2}-20 x-7}$$

Step-by-Step Solution

Answer

The domain of the given expression is \(-1/3 < x < 7\) and \(x > 7\).

Step 1: Write down the inequality

The inequality to solve to find the domain is \(3x^{2}-20x-7 \geq 0\).

Step 2: Factor the quadratic

Factor the quadratic to make it easier to find the roots. Factoring gives \((3x + 1)(x - 7) \geq 0\).

Step 3: Find the roots

Set the factors to zero and solve for \(x\) to find the roots. The roots are at \(x = -1/3\) and \(x = 7\).

Step 4: Identify the intervals

The intervals are therefore \((-\infty, -1/3)\), \((-1/3, 7)\) and \((7, \infty)\)

Step 5: Test the intervals

Choose a test point from each interval, plug it into the inequality (in factored form), and check the sign. From this, the intervals where the inequality is true are \((-1/3, 7)\) and \((7, \infty)\)

Step 6: Write the domain

Putting it all together, this gives us the domain of \(x\), which are \(-1/3 < x < 7\) and \(x > 7\)

Key Concepts

Quartic Root of Function Domain

Understanding the domain of a function is crucial because it tells us the set of all possible inputs (or 'x' values) for which the function is defined. A quartic root, which is represented as \( \sqrt[4]{f(x)} \) in mathematical terms, introduces a restriction to the domain: the function underneath the quartic root, \( f(x) \) in this case, must be greater than or equal to zero. This is because in real numbers, you cannot take an even root, including the quartic root, of a negative number without obtaining an imaginary result.

In the given exercise, we are asked to find the domain of the expression \( \sqrt[4]{3x^2-20x-7} \). To ensure that the value under the root is non-negative, we set up the inequality \( 3x^2-20x-7 \geq 0 \). Solving this inequality will provide us with the intervals for the domain of \( x \). Identifying the correct domain is fundamental to avoid errors in further calculations involving the function.

Factoring Quadratic Equations

Factoring quadratic equations is an essential technique when working with inequalities to find domains of functions, among other applications. A quadratic equation generally takes the form \( ax^2 + bx + c = 0 \) and can often be factored into two binomials. Factoring simplifies solving the equation or inequality by breaking it down into its root components.

In the solution for the above exercise, we factor the quadratic \( 3x^2-20x-7 \) to find \( (3x + 1)(x - 7) \). Each factor represents a potential root of the equation when set equal to zero. By factoring, we can easily identify the zeroes of the quadratic component of the expression, which are critical to determining where the function changes sign, and hence, where it satisfies the inequality needed to find the domain.

Inequalities and Intervals

Inequalities like \( 3x^2-20x-7 \geq 0 \) are solved not just to find single values but to identify entire intervals of numbers that satisfy the condition. These intervals form part of the solution because a quadratic inequality will be true over a range of 'x' values rather than at just a single point. After factoring the quadratic equation and finding the roots, we use these roots to divide the number line into intervals.

In our exercise, the roots \( x = -1/3 \) and \( x = 7 \) divide the number line into three intervals: \( (-\infty, -1/3) \) where the function is negative, \( (-1/3, 7) \) where the function is positive, and \( (7, \infty) \) where the function is again positive. By testing points from each interval in the original inequality, we confirm which intervals make the inequality true. These two positive intervals, \( (-1/3, 7) \) and \( (7, \infty) \) make up our solution for the domain of the function in question. Understanding intervals in the context of inequalities is pivotal for interpreting and solving many problems in algebra.