Find the domain of the function.
[tex]f(x)=\sqrt{\frac{3}{x+7}}[/tex]
Write your answer as an interval or union of intervals.
Answer (2)
The domain of the function f ( x ) = x + 7 3 is all real numbers greater than -7. In interval notation, this is expressed as ( − 7 , ∞ ) . This ensures that the expression under the square root is defined and non-negative.
;
The function is defined when the expression inside the square root is non-negative.
Set up the inequality 0"> x + 7 3 > 0 .
Solve the inequality to find -7"> x > − 7 .
Express the solution in interval notation: ( − 7 , ∞ ) .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = x + 7 3 . The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a square root function, which means the expression inside the square root must be non-negative. Additionally, we have a fraction, so the denominator cannot be zero.
Setting up the Inequality For the function to be defined, the expression inside the square root must be greater than or equal to zero: x + 7 3 ≥ 0 Since the numerator is a positive constant (3), the fraction will be positive if and only if the denominator is positive. Therefore, we need to find the values of x for which: 0"> x + 7 > 0 Note that we use "> > and not ≥ because the denominator cannot be zero.
Solving the Inequality Now, we solve the inequality for x : 0"> x + 7 > 0 Subtract 7 from both sides: -7"> x > − 7 This means that the function is defined for all x values greater than -7.
Expressing the Solution as an Interval We express the solution as an interval. Since x must be greater than -7, the domain of the function is all real numbers greater than -7, not including -7. In interval notation, this is written as: ( − 7 , ∞ )
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if x represents the time in seconds since a chemical reaction started, and f ( x ) represents the concentration of a product, then finding the domain of f ( x ) tells us for what times the concentration is a valid, non-negative real number. If the function is f ( x ) = x + 7 3 , then the reaction is only valid for -7"> x > − 7 , which practically means we only care about x ≥ 0 since time cannot be negative. This ensures that we are dealing with meaningful, real-world values.
