Problem 62 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\frac{x^{2}}{\sqrt{6-x}} $$

Step-by-Step Solution

Answer

The domain of \( f(x) \) is \((-\infty, 6)\).

Step 1: Identify Domain Restrictions

The function \( f(x) = \frac{x^2}{\sqrt{6-x}} \) will be undefined at any point where the denominator \( \sqrt{6-x} \) is zero or negative. Therefore, we must find when the expression under the square root is non-negative.

Step 2: Set Up Inequality

The expression \( 6-x \) must be greater than zero because the square root works only with non-negative expressions. Set up the inequality: \( 6-x > 0 \).

Step 3: Solve Inequality

Solve the inequality \( 6-x > 0 \) to find the allowed values for \( x \). By subtracting \( 6 \) from both sides, we have \( -x > -6 \). By dividing both sides of the inequality by \( -1 \), and flipping the inequality sign, we get \( x < 6 \).

Step 4: Domain Conclusion

Since \( x \) must be less than 6 and there are no other restrictions from the numerator \( x^2 \), the domain of the function is all real numbers less than 6. Expressed in interval notation, the domain is \((-\infty, 6)\).

Key Concepts

Inequalities

Inequalities are a type of mathematical expression that involve comparing the relative size of two values. Instead of showing equality, inequalities use signs such as \( > \), \( < \), \( \geq \), and \( \leq \). These symbols show whether one side is greater or smaller than the other.In the context of functions, inequalities often help us determine valid inputs by defining conditions under which the function is defined. For instance, in the expression \( 6-x > 0 \), we establish a permissible range for \( x \). By solving this inequality, we rearrange the terms to find where it holds true. Here, \( 6-x > 0 \) means \( x < 6 \). This solution process ensures that the function's operation remains valid by preventing undefined calculations, like a square root of a negative number in this case.

Square Root Expressions

Square root expressions are prevalent in mathematics, especially when delving into functions that involve some form of the radical notation like \( \sqrt{6-x} \). These expressions hold specific rules and restrictions mainly due to the nature of square roots themselves.

• The key rule to remember is that the expression inside a square root must be non-negative, that is, zero or positive.
• If a number under the root is negative, the root isn't defined in the realm of real numbers.

By focusing on ensuring that the expression under the square root in our function, \( 6-x \), is positive, we maintain the function's real number domain. Solving \( 6-x > 0 \) becomes essential to hint where the function is calculable. For the function \( f(x) = \frac{x^2}{\sqrt{6-x}} \), having a non-negative root expression ensures the function evaluates properly, maintaining its validity across its domain.

Function Restrictions

Functions often come with restrictions that dictate where they can properly operate. These restrictions are essential, as they avoid undefined behavior such as division by zero or taking square roots of negative numbers.To identify these restrictions, look at different parts of the function:

• Denominators: Functions with fractions have to avoid dividing by zero.
• Radicals (like square roots): The expression inside must be non-negative to keep the function in real numbers.

For our function, \( f(x)=\frac{x^{2}}{\sqrt{6-x}} \), the restriction stems from the square root in the denominator. It requires \( 6-x > 0 \) to make \( \sqrt{6-x} \) defined in real numbers. There are no additional restrictions contributed by the numerator, which is simply \( x^2 \). By identifying these restrictions, we specify a valid input range for \( x \), thus defining the domain of the function.