5 Domain Word Problems

Introduction to Domain Word Problems

Domain word problems are a crucial part of mathematics, particularly in algebra and functions. These problems require students to understand the concept of domain, which refers to the set of all possible input values for a function. In this article, we will explore five different domain word problems, along with their solutions and explanations.

Problem 1: Finding the Domain of a Square Root Function

The function f(x) = \sqrt{x-3} represents the square root of the difference between x and 3. To find the domain of this function, we need to consider the values of x that make the expression inside the square root non-negative.

💡 Note: The expression inside a square root must be non-negative, as the square root of a negative number is undefined in real numbers.

The expression x-3 \geq 0 must hold true, which simplifies to x \geq 3. Therefore, the domain of the function f(x) is all real numbers greater than or equal to 3.

Problem 2: Determining the Domain of a Rational Function

The rational function f(x) = \frac{x+2}{x-2} has a restriction on its domain due to the denominator. We cannot have the denominator equal to zero, as division by zero is undefined. The equation x-2 \neq 0 implies that x \neq 2. Hence, the domain of the function f(x) is all real numbers except 2.

Problem 3: Finding the Domain of a Function with Absolute Value

Consider the function f(x) = |x-5| + 3. Since the absolute value of any real number is non-negative, the expression |x-5| is always non-negative. Adding 3 to this expression does not impose any restrictions on the domain. Therefore, the domain of the function f(x) is all real numbers.

Problem 4: Domain of a Function Involving a Quadratic Expression

The function f(x) = \frac{1}{x^2-4} has a restriction on its domain due to the denominator. We cannot have the denominator equal to zero, so we must find the values of x that make the quadratic expression x^2-4 equal to zero. Solving the equation x^2-4 = 0, we get x = \pm 2. Hence, the domain of the function f(x) is all real numbers except -2 and 2.

Problem 5: Domain of a Function with a Radical Expression

The function f(x) = \sqrt[3]{x-1} involves a cube root, which is defined for all real numbers. However, we still need to consider the expression inside the cube root. Since the cube root is defined for all real numbers, the domain of the function f(x) is all real numbers.

Function	Domain
$f(x) = \sqrt{x-3}$	$x \geq 3$
$f(x) = \frac{x+2}{x-2}$	$x \neq 2$
$f(x) = |x-5| + 3$	All real numbers
$f(x) = \frac{1}{x^2-4}$	$x \neq \pm 2$
$f(x) = \sqrt[3]{x-1}$	All real numbers

In summary, domain word problems require us to consider the restrictions on the input values of a function. By analyzing the function and its components, we can determine the set of all possible input values, which is the domain of the function. Key points to remember include the properties of square roots, rational functions, absolute value, and radical expressions.