$f(x)=\sqrt{x+4}$
a) Find the domain
Answer (1)
The domain of f ( x ) = x + 4 requires x + 4 ≥ 0 .
Solve the inequality x + 4 ≥ 0 for x .
The solution is x ≥ − 4 .
Express the domain in interval notation: [ − 4 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x + 4 and asked to find its domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Setting up the Inequality Since we have a square root, the expression inside the square root must be greater than or equal to zero. This is because the square root of a negative number is not a real number. So, we need to solve the inequality: x + 4 ≥ 0
Solving the Inequality To solve the inequality, we subtract 4 from both sides: x + 4 − 4 ≥ 0 − 4 x ≥ − 4
Expressing the Solution in Interval Notation This means that the domain of the function is all real numbers x such that x is greater than or equal to -4. In interval notation, this is written as [ − 4 , ∞ ) .
Final Answer Therefore, the domain of the function f ( x ) = x + 4 is [ − 4 , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if x represents the time in seconds after an event, and f ( x ) represents the distance an object travels, then the domain of f ( x ) tells us the valid range of times for which the distance function is meaningful. In this case, since x ≥ − 4 , it means we can only consider times from -4 seconds onwards. This could represent a scenario where we start measuring time 4 seconds before the actual event occurs. Knowing the domain ensures we're working with realistic and meaningful values.
