Problem 64
Question
Find the domain of each function. $$ f(x)=\sqrt{x+2} $$
Step-by-Step Solution
Verified Answer
The domain of the function \(f(x) = \sqrt{x + 2}\) is \(x \geq -2\).
1Step 1: Set up the inequality
Set up the inequality to find the values of x that will make the expression under the square root greater than or equal to zero: \(x + 2 \geq 0\)
2Step 2: Solve the inequality
Subtract 2 from both sides of the inequality to solve for x: \(x \geq -2\)
3Step 3: State the domain
The solution to the inequality is x \geq -2. Therefore, the domain of the function \(f(x) = \sqrt{x + 2}\) is all real numbers greater than or equal to -2.
Key Concepts
Domain of Square Root FunctionSolving InequalitiesRadical FunctionsCollege Algebra
Domain of Square Root Function
The domain of any function is the set of input values, or 'x' values, for which the function is defined. Specifically for a square root function, such as \( f(x) = \sqrt{x + 2} \) that we're considering, the expression under the square root (also known as the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number, and most of the functions we deal with in college algebra are real-valued.
When finding the domain of \( f(x) = \sqrt{x + 2} \) we set the radicand \( x + 2 \) to be greater than or equal to zero, leading to the inequality \( x + 2 \geq 0 \). This gives us all x values that make the function real and hence definable. Consequently, the domain here is all real numbers \( x \geq -2 \). Conveying this in interval notation, we would use \( [-2, \infty) \) to show that the function includes -2 and all numbers greater than -2.
When finding the domain of \( f(x) = \sqrt{x + 2} \) we set the radicand \( x + 2 \) to be greater than or equal to zero, leading to the inequality \( x + 2 \geq 0 \). This gives us all x values that make the function real and hence definable. Consequently, the domain here is all real numbers \( x \geq -2 \). Conveying this in interval notation, we would use \( [-2, \infty) \) to show that the function includes -2 and all numbers greater than -2.
Solving Inequalities
Solving inequalities is a fundamental part of understanding functions and their domains in college algebra. Unlike equations, inequalities do not show exact values but rather a range of possible values that satisfy the condition. In our exercise, \( x + 2 \geq 0 \) is the inequality in question.
To solve for \( x \) in this inequality, we perform basic algebraic operations that maintain the inequality's direction, unless we multiply or divide by a negative number, in which case the direction of the inequality would flip. In the given problem, we simply subtract 2 from both sides, which maintains the inequality and isolates \( x \) on one side, giving us the solution \( x \geq -2 \). This simple act of subtracting 2 demonstrates the core principle of keeping operations balanced when solving inequalities.
To solve for \( x \) in this inequality, we perform basic algebraic operations that maintain the inequality's direction, unless we multiply or divide by a negative number, in which case the direction of the inequality would flip. In the given problem, we simply subtract 2 from both sides, which maintains the inequality and isolates \( x \) on one side, giving us the solution \( x \geq -2 \). This simple act of subtracting 2 demonstrates the core principle of keeping operations balanced when solving inequalities.
Radical Functions
Radical functions contain roots, such as the square root, within their formula. The square root function is a type of radical function with the index of two, meaning it looks for a number that, when multiplied by itself, will yield the radicand. Due to the nature of square roots, the radicand must be a non-negative number to stay within the real number system.
Consideration of domain is particularly important for radical functions, as it directly influences the type of inputs that can be placed into the function. For the square root function \( f(x) = \sqrt{x + 2} \), the radicand \( x + 2 \) sets the stage for the domain. Ensuring the radicand is non-negative allows us to utilize radical functions fully and properly within the realm of college algebra.
Consideration of domain is particularly important for radical functions, as it directly influences the type of inputs that can be placed into the function. For the square root function \( f(x) = \sqrt{x + 2} \), the radicand \( x + 2 \) sets the stage for the domain. Ensuring the radicand is non-negative allows us to utilize radical functions fully and properly within the realm of college algebra.
College Algebra
College algebra serves as the backbone for a variety of disciplines, including science, engineering, healthcare, economics, and social sciences. Mastering the concepts of domain, radical functions, and solving inequalities are crucial components of this subject. Algebraic thinking allows us to model the real world and solve problems.
Understanding the domain of functions helps students to graph functions accurately and to anticipate the possible outputs. As in the previous examples, knowing how to determine the domain ensures that the values we utilize make sense within the context of the function in question, such as when dealing with square root and radical functions. Consequently, a significant aspect of college algebra is developing the ability to discern constraints on function inputs and interpret them within a broader mathematical context.
Understanding the domain of functions helps students to graph functions accurately and to anticipate the possible outputs. As in the previous examples, knowing how to determine the domain ensures that the values we utilize make sense within the context of the function in question, such as when dealing with square root and radical functions. Consequently, a significant aspect of college algebra is developing the ability to discern constraints on function inputs and interpret them within a broader mathematical context.
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