Problem 11 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\sqrt{x}+\sqrt{1-x} $$

Step-by-Step Solution

Answer

The domain of the function is \([0, 1]\).

Step 1: Understand the Function

The function is given as \( f(x) = \sqrt{x} + \sqrt{1-x} \). It includes two square root expressions: \( \sqrt{x} \) and \( \sqrt{1-x} \). Since square roots are involved, both expressions under the square roots must be non-negative.

Step 2: Determine the Domain for \( \sqrt{x} \)

The expression \( \sqrt{x} \) requires \( x \geq 0 \). This means that in order for \( \sqrt{x} \) to exist, the values of \( x \) must be greater than or equal to zero.

Step 3: Determine the Domain for \( \sqrt{1-x} \)

The expression \( \sqrt{1-x} \) requires \( 1-x \geq 0 \), which simplifies to \( x \leq 1 \). This condition ensures the values under the square root are non-negative.

Step 4: Combine the Conditions

Considering \( x \geq 0 \) from Step 2 and \( x \leq 1 \) from Step 3, we must find the values of \( x \) that satisfy both conditions. This results in the combined condition of \( 0 \leq x \leq 1 \).

Step 5: Write the Domain

The domain of the function \( f(x) = \sqrt{x} + \sqrt{1-x} \) is all values of \( x \) that satisfy both conditions established. Therefore, the domain is the interval \([0, 1]\).

Key Concepts

Square Root Function

Let's delve into the square root function, which is a fundamental mathematical concept. A square root function is any function that involves the square root of a variable. The general form can be expressed as \( f(x) = \sqrt{x} \). The crucial point to understand about square root functions is that they are defined only for non-negative values.
For example, \( \sqrt{x} \) is defined when \( x \geq 0 \), because you cannot take the square root of a negative number and obtain a real number result. This property affects the domain of any function involving square roots, requiring us to analyze the interior terms for non-negativity.
This brings us to our exercise:

• For \( \sqrt{x} \), the condition is that \( x \) must be 0 or greater.
• For \( \sqrt{1-x} \), \( 1-x \) must be non-negative, simplifying to \( x \leq 1 \).

Thus, these conditions help us determine the range of acceptable \( x \) values for the function to be defined.

Interval Notation

In mathematics, interval notation is a succinct way of expressing a range of values, especially useful when defining domains. When we want to include a continuous set or interval of numbers, we use interval notation to clearly show which numbers are included.
Let's break down some basics:

• Square brackets \( [ ] \) denote that the endpoint is included in the interval. For example, \([a, b]\) means all numbers from \(a\) to \(b\) inclusive.
• Parentheses \( ( ) \) indicate that the endpoint is not included. For example, \((a, b)\) means all numbers between \(a\) and \(b\), not including \(a\) and \(b\) themselves.

Referring back to our function, with domain \( 0 \leq x \leq 1 \), we represent this with interval notation as \[ [0, 1] \], including both endpoints. This precisely communicates the set of \( x \) values where the function \( f(x) \) is valid.

Function Analysis

Function analysis involves several critical tasks such as identifying domains, ranges, and specifics of the behavior and characteristics of mathematical functions. For the function \( f(x) = \sqrt{x} + \sqrt{1-x} \), our primary analysis task is to determine its domain.
This requires checking the constraints each part of the function imposes:

• \( \sqrt{x} \) indicates \( x \) must be non-negative, hence \( x \geq 0 \).
• \( \sqrt{1-x} \) requires \( 1-x \geq 0 \), which simplifies to \( x \leq 1 \).

Only values of \( x \) that satisfy both \( x \geq 0 \) and \( x \leq 1 \) will make the entire function defined, forming the domain [0, 1]. This thorough function analysis ensures correct understanding and application of the square root constraints, within the given context.