Problem 11 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\sqrt{x^{2}+4} $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, \infty)\).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{x^2 + 4} \). This is a square root function, which typically requires the expression inside the square root to be greater than or equal to zero.

Step 2: Analyze the Function's Expression

For square root functions, we need to ensure that the expression under the square root is non-negative: \( x^2 + 4 \geq 0 \). Since the term \( x^2 \) is always non-negative for all real numbers, adding 4 ensures the sum \( x^2 + 4 \) is always positive (greater than 4).

Step 3: Determine the Domain

Given that \( x^2 + 4 \) is always positive regardless of the real number we substitute for \( x \), the domain of the function is all real numbers. Thus, the domain in interval notation is \( (-\infty, \infty) \).

Key Concepts

Square Root Function

The square root function is a mathematical operation in which a number, known as the radicand, is divided to find a value that, when multiplied by itself, gives the original number. In simpler terms, it's like asking "what number squared gives me this?" Square root functions are customarily written using the square root symbol, \( \sqrt{} \).
For a function like \( f(x) = \sqrt{x^2 + 4} \), this means you are looking for values of \( x \) that will keep the value inside the square root non-negative.
This particular function includes \( x^2 + 4 \) under the square root sign, ensuring it is always positive due to the properties of squared numbers and addition with a positive constant. Therefore, this function can accept any real number as \( x \), which is slightly unusual as square-root functions often restrict \( x \) to prevent getting negative radicands.
When simplified, this gives us a broad domain of real numbers!

Interval Notation

Interval notation is a concise way of describing ranges of values. It is especially useful in mathematics for denoting domains and ranges of functions.

• \([a, b]\): This represents all numbers from \(a\) to \(b\), including both \(a\) and \(b\).
• \( (a, b) \): This includes all numbers between \(a\) and \(b\), but not \(a\) or \(b\) themselves.
• \( [a, b) \) or \( (a, b] \) : Indicates a mix where one endpoint is included and the other is not.
• \( (-\infty, b) \) or \( (a, \infty) \): These notations describe numbers extending indefinitely to the left or right on the number line, with infinity denoting unboundedness.

For the function \( f(x) = \sqrt{x^2 + 4} \), since \( x^2 + 4 \) is always positive or greater for any real number, all real numbers are included in the domain. Hence, the domain in interval notation is \( (-\infty, \infty) \). This denotes that any real number can be input into the function.

Real Numbers

Real numbers are the backbone of our number system, encompassing almost every number you encounter in everyday life. They include rational numbers (like 1/2, 4, and -9) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)).

• Rational Numbers: These can be expressed as fractions of integers. For example, 3.75 and 5/8 are both rational.
• Irrational Numbers: These cannot be expressed as simple fractions. Examples include the square root of any non-perfect square, or numbers like \( \pi \).
• Whole Numbers: These are non-negative numbers without fractions or decimals, such as 0, 1, 2, etc.
• Integers: Whole numbers that also include negatives, such as -2, 0, and 7.

Real numbers constitute a continuous range, meaning you won't jump to another number without meeting every number in between first. This property of real numbers makes them incredibly useful for defining domains in functions, particularly when the function requires a large or unrestricted domain such as \( f(x) = \sqrt{x^2 + 4} \). In this case, any real number works as an input, leading to the domain being expressed as \( (-\infty, \infty) \).