Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does the domain of the function \(f(x)=\sqrt{1+x}\) include \(x=-2 ?\)

Step-by-Step Solution

Answer

No, the domain of the function \(f(x)=\sqrt{1+x}\) does not include x=-2.

Step 1: Understand the function's domain constraints

For the function \(f(x)=\sqrt{1+x}\), we can see that it is a square root function which only yields real numbers when the number inside the square root is equal to or greater than 0. Therefore, the domain of \(f(x)\) is \(x \geq -1\).

Step 2: Evaluate if x=-2 is within the function's domain

Now, let's check if x=-2 is within the function's domain. Based on the domain's constraints found in step 1, we see that x=-2 is less than -1. Therefore, it is not within the function's domain.

Key Concepts

Square Root Function

The square root function is a type of radical function that finds the square root of a given number. In its most basic form, it is represented as \( f(x) = \sqrt{x} \). The square root symbol \( \sqrt{} \) indicates that we need to find a number which, when multiplied by itself, results in the original number under the square root.
The defining feature of this function is that it only produces real outputs if the input is non-negative. This is because the square root of a negative number does not exist within the set of real numbers, but rather in the set of imaginary numbers.
For example, the square root of 4 is 2, because 2 multiplied by itself returns 4. It's important to understand this when dealing with functions that incorporate square roots.

• The input values must make the expression inside the square root non-negative.
• This leads to specific restrictions known as domain constraints.

Domain Constraints

Domain constraints refer to the limitations on the set of possible input values, or the domain, for a function. Especially with square root functions, these constraints ensure that the output remains within a set of real numbers.
In the function \( f(x) = \sqrt{1+x} \), the expression inside the square root must be equal to or greater than zero for the function to yield real numbers. In mathematical terms, this is expressed through the inequality \( 1 + x \geq 0 \).
Solving this inequality involves isolating x:

• Subtract 1 from both sides to find \( x \geq -1 \).

Hence, the domain of \( f(x) = \sqrt{1+x} \) is all numbers \( x \) such that \( x \geq -1 \). Any value of \( x \) less than -1 is outside the domain, as it would result in trying to find the square root of a negative number, which is not defined in real numbers.

Real Numbers

Real numbers include all the numbers on the real number line. This means positive numbers, negative numbers, fractions, and decimals—even zero. Importantly, real numbers do not include imaginary or complex numbers, which are numbers involving the square root of a negative number.
For functions like \( f(x) = \sqrt{1+x} \), understanding real numbers is crucial, as the square root must yield a real number to be valid.

• We check if the expression in the square root is non-negative.
• If it is, the function will yield a real output.

In practical terms, whenever a function is expected to provide real number outputs, ensuring the domain meets the condition of non-negative inputs inside square roots is necessary.

Inequalities

Inequalities are mathematical expressions that indicate the relative size or order of two values. They are often used to express domain constraints. In the context of \( f(x) = \sqrt{1+x} \), the inequality \( 1+x \geq 0 \) ensures we differentiate between values that provide valid, real number outputs from those that don't.
Such inequalities can be solved to understand what input values, or x, satisfy the condition. Here are key aspects:

• Simplify the inequality to isolate x.
• This means solving for x in \( 1+x \geq 0 \), resulting in \( x \geq -1 \).
• The solution tells us all possible x values in the function's domain.

Inequalities help establish boundaries necessary for functions to operate over real numbers properly.