Domain of the Square Root Function f(x) = √x
The domain of the function f(x) = √x is the set of all real numbers x for which √x is defined. The square root function is defined only for non-negative numbers, including zero.
Interval Notation
In interval notation, the domain of f(x) = √x is:
[0, ∞)
Set-Builder Notation
In set-builder notation, the domain is expressed as:
{ x ∈ ℝ : x ≥ 0 }
Inequality Form
In inequality form, the domain is simply:
x ≥ 0
Explanation
1. Interval Notation: [0, ∞) means that x can be any real number greater than or equal to 0. The square bracket [ indicates that 0 is included in the domain.
2. Set-Builder Notation: { x ∈ ℝ : x ≥ 0 } means that x is an element of the set of real numbers (ℝ) and must satisfy the condition x ≥ 0.
3. Inequality Form: x ≥ 0 is a straightforward way to express that x can be any real number that is greater than or equal to 0.
In all these forms, the idea is the same: the function √x is defined only when x is a non-negative real number.
Finding the Domain of f(x) = √(x – 1)
Why the Inequality?
The function we’re looking at is f(x) = √(x – 1). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, x – 1, must be greater than zero for the function to be defined.
Setting Up the Inequality
To ensure that the square root is defined, we set up the inequality:
x – 1 > 0
This inequality is saying that x – 1 must be greater than zero for √(x – 1) to be a real number.
Solving for x
To find the values of x that satisfy this inequality, we isolate x on one side. We do this by adding 1 to both sides of the inequality:
x > 1
Now, x is isolated, and we find that x must be greater than 1 for the function f(x) = √(x – 1) to be defined.
Finding the Domain of f(x) = √(x – 1), Continued
Converting to Other Notations
Interval Notation
Start with the inequality x > 1. In interval notation, we use parentheses to indicate that a number is not included. Since x > 1 (and 1 is not included), we start the interval with a round parenthesis: (1, ∞).
Set-Builder Notation
Start with the inequality x > 1. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x > 1 } tells us that x is an element of the set of all real numbers and must be greater than 1.
Inequality Form
The inequality form x > 1 is the simplest way to express the domain. It directly comes from our inequality solution.
Finding the Domain of f(x) = √(x + 2)
Finding the Domain through Inequality
1. Why the Inequality? The function we’re looking at is f(x) = √(x + 2). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, x + 2, must be greater than zero for the function to be defined.
2. Setting Up the Inequality: To ensure that the square root is defined, we set up the inequality:
x + 2 > 0
3. Solving for x: To find the values of x that satisfy this inequality, we isolate x on one side by subtracting 2 from both sides:
x > -2
Converting to Other Notations
1. Interval Notation: Start with the inequality x > -2. In interval notation, we use parentheses to indicate that a number is not included. Since x > -2 (and -2 is not included), we start the interval with a round parenthesis: (-2, ∞).
2. Set-Builder Notation: Start with the inequality x > -2. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x > -2 } tells us that x is an element of the set of all real numbers and must be greater than -2.
Finding the Domain of f(x) = √(2x + 1)
Finding the Domain through Inequality
1. Why the Inequality? The function we’re looking at is f(x) = √(2x + 1). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, 2x + 1, must be greater than zero for the function to be defined.
2. Setting Up the Inequality: To ensure that the square root is defined, we set up the inequality:
2x + 1 > 0
3. Solving for x: To find the values of x that satisfy this inequality, we isolate x on one side by subtracting 1 from both sides and then dividing by 2:
x > -1/2
Converting to Other Notations
1. Interval Notation: Start with the inequality x > -1/2. In interval notation, we use parentheses to indicate that a number is not included. Since x > -1/2 (and -1/2 is not included), we start the interval with a round parenthesis: (-1/2, ∞).
2. Set-Builder Notation: Start with the inequality x > -1/2. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x > -1/2 } tells us that x is an element of the set of all real numbers and must be greater than -1/2.
Finding the Domain of f(x) = √(-x)
Finding the Domain through Inequality
1. Why the Inequality? The function we’re looking at is f(x) = √(-x). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, -x, must be greater than zero for the function to be defined.
2. Setting Up the Inequality: To ensure that the square root is defined, we set up the inequality:
-x > 0
3. Solving for x: To find the values of x that satisfy this inequality, we isolate x on one side by multiplying both sides by -1 and reversing the inequality sign:
x < 0
Converting to Other Notations
1. Interval Notation: Start with the inequality x < 0. In interval notation, we use parentheses to indicate that a number is not included. Since x < 0 (and 0 is not included), we start the interval with a round parenthesis: (-∞, 0).
2. Set-Builder Notation: Start with the inequality x < 0. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x < 0 } tells us that x is an element of the set of all real numbers and must be less than 0.
Finding the Domain of f(x) = √(-x + 2)
Finding the Domain through Inequality
1. Why the Inequality? The function we’re looking at is f(x) = √(-x + 2). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, -x + 2, must be greater than zero for the function to be defined.
2. Setting Up the Inequality: To ensure that the square root is defined, we set up the inequality:
-x + 2 > 0
3. Solving for x: To find the values of x that satisfy this inequality, we isolate x on one side by adding x to both sides and then subtracting 2:
x < 2
Converting to Other Notations
1. Interval Notation: Start with the inequality x < 2. In interval notation, we use parentheses to indicate that a number is not included. Since x < 2 (and 2 is not included), we start the interval with a round parenthesis: (-∞, 2).
2. Set-Builder Notation: Start with the inequality x < 2. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x < 2 } tells us that x is an element of the set of all real numbers and must be less than 2.
Finding the Domain of f(x) = √(2x – 1)
Finding the Domain through Inequality
1. Why the Inequality? The function we’re looking at is f(x) = √(2x – 1). The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, 2x – 1, must be greater than zero for the function to be defined.
2. Setting Up the Inequality: To ensure that the square root is defined, we set up the inequality:
2x – 1 > 0
3. Solving for x: To find the values of x that satisfy this inequality, we isolate x on one side by adding 1 to both sides and then dividing by 2:
x > 1/2
Converting to Other Notations
1. Interval Notation: Start with the inequality x > 1/2. In interval notation, we use parentheses to indicate that a number is not included. Since x > 1/2 (and 1/2 is not included), we start the interval with a round parenthesis: (1/2, ∞).
2. Set-Builder Notation: Start with the inequality x > 1/2. In set-builder notation, we describe the set of all numbers that satisfy the inequality. The notation { x ∈ ℝ : x > 1/2 } tells us that x is an element of the set of all real numbers and must be greater than 1/2.
Finding the Domain of f(x) = √(-2x – 3)
Steps for Solving the Inequality
1. Setting Up the Inequality: The expression inside the square root, -2x – 3, must be greater than or equal to zero for the function to be defined. So, we set up the inequality:
-2x – 3 ≥ 0
2. Isolating x: To find the values of x that satisfy this inequality, we’ll isolate x on one side.
– First, add 3 to both sides:
-2x ≥ 3
– Next, divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number:
x ≤ -3/2
Now, x is isolated, and we find that x must be less than or equal to -3/2 for the function f(x) = √(-2x – 3) to be defined.
Finding the Domain of \( f(x) = \sqrt{\frac{1}{2}x – 1} \)
Steps for Solving the Inequality
1. Setting Up the Inequality: The expression inside the square root, \( \frac{1}{2}x – 1 \), must be greater than or equal to zero for the function to be defined. So, we set up the inequality:
\( \frac{1}{2}x – 1 \geq 0 \)
2. Isolating \( x \): To find the values of \( x \) that satisfy this inequality, we’ll isolate \( x \) on one side.
– First, add 1 to both sides:
\( \frac{1}{2}x \geq 1 \)
– Next, multiply both sides by 2 to get rid of the fraction:
\( x \geq 2 \)
Now, \( x \) is isolated, and we find that \( x \) must be greater than or equal to 2 for the function \( f(x) = \sqrt{\frac{1}{2}x – 1} \) to be defined.