Problem 26 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\) (b) \(g(x)=\sqrt{x+2}-\sqrt{x-1}\)

Step-by-Step Solution

Answer

The domain for function \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\) is all real numbers except -2 and 1. The domain for function \(g(x)=\sqrt{x+2}-\sqrt{x-1}\) is all real numbers greater than or equal to 1.

Step 1: Determine the Domain of Function (a)

For the function \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\), the denominator cannot be zero (undefined) in a fraction. So, we'll set each denominator not equal to zero and solve for \(x\). \n\n- For \(x+2\neq0\), we find \(x\neq-2\). \n- For \(x-1\neq0\), we find \(x\neq1\). Thus, the domain for function \(f\) is all real numbers except -2 and 1.

Step 2: Determine the Domain of Function (b)

For the function \(g(x)=\sqrt{x+2}-\sqrt{x-1}\), note that we cannot take the square root of a negative number. Therefore, the domain is where each square root expression is not negative. So, we'll set each expression inside the square root not less than zero and solve for \(x\). \n\n- For \(x+2\geq0\), we find \(x\geq-2\). \n- For \(x-1\geq0\), we find \(x\geq1\). Therefore, the domain of function \(g\) is all real numbers greater than or equal to 1, because this satisfies both conditions.

Key Concepts

Determining Function Domain

The domain of a function comprises all the possible inputs (typically represented as 'x' values) that the function can accept without leading to any undefined or non-real results. To determine the domain, we assess the function's rules and restrictions like division by zero and root operations on negative numbers.

Let's take an example. For the function \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\), we need to find the values for 'x' that will not result in division by zero, since dividing by zero is undefined in mathematics. By excluding the values that equate the denominator to zero (\(x=-2\) and \(x=1\)), we establish the domain of \(f(x)\) as all real numbers except -2 and 1. This step-by-step approach ensures we find the largest possible domain for a given function.

Undefined Expressions in Functions

Undefined expressions in functions primarily occur when we violate basic mathematical rules, such as division by zero or taking roots of negative numbers when dealing with real numbers. In the case of the function \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\), the denominators \(x+2\) and \(x-1\) lead to undefined expressions if they equal zero.

When finding the domain, we consequently rule out the \(x\) values that would cause this, which are \(x=-2\) and \(x=1\). Allowing these values would mean the function could not be evaluated, thus they must be excluded from the domain. This is a critical step in ensuring the function is well-defined over its entire domain.

Square Root Domain Restrictions

When a function includes square root expressions, such as \(g(x)=\sqrt{x+2}-\sqrt{x-1}\), the domain is restricted to prevent square roots of negative numbers, which are undefined in the real number system. This restriction is fundamental for determining the domain of functions with square root terms.

To find the allowable 'x' values for \(g(x)\), we set the expressions under each square root to be greater than or equal to zero: \(x+2\geq0\) and \(x-1\geq0\), which results in \(x\geq-2\) and \(x\geq1\), respectively. Since both conditions need to be satisfied, the domain is all real numbers \(x\) greater than or equal to 1. This illustrates how restrictions for each term in a function can affect the overall domain.

Real Number Domain Constraints

The domain constraints for real numbers revolve around maintaining real-number outputs for any input within the domain. This means excluding inputs that lead to non-real outputs, such as square roots of negative numbers, and maintaining adherence to defined operations like avoiding division by zero.

Understanding these principles helps in accurately sketching domain constraints for real-valued functions. For instance, the domain found for \(g(x)=\sqrt{x+2}-\sqrt{x-1}\) is all real numbers greater than or equal to 1 because this ensures all outputs remain in the realm of real numbers. Efficiently determining these constraints is a key skill in mastering function domain identification.