Find the domain of the function.
[tex]f(x)=\frac{10+x}{\sqrt{-6+x}}[/tex]
Write your answer as an interval or union of intervals.
Answer (2)
The domain of the function f ( x ) = − 6 + x 10 + x is ( 6 , ∞ ) . This means that the function is defined for all values greater than 6. Inputs less than or equal to 6 are not valid since the square root in the denominator would be zero or negative.
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The function is defined when the expression inside the square root is strictly positive: 0"> − 6 + x > 0 .
Solve the inequality: 6"> x > 6 .
Express the solution in interval notation: ( 6 , ∞ ) .
The domain of the function is ( 6 , ∞ ) .
Explanation
Analyze the function We are given the function f ( x ) = − 6 + x 10 + x and we need 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. In this case, we have a rational function with a square root in the denominator.
Set up the inequality For the function to be defined, the expression inside the square root must be strictly positive, because we cannot take the square root of a negative number, and the denominator cannot be zero. Therefore, we must have 0"> − 6 + x > 0 .
Solve the inequality To solve the inequality 0"> − 6 + x > 0 , we add 6 to both sides: 6"> x > 6
Express the domain in interval notation This means that the function is defined for all x greater than 6. In interval notation, this is expressed as ( 6 , ∞ ) .
State the final answer Therefore, the domain of the function f ( x ) = − 6 + x 10 + x is ( 6 , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, if f ( x ) represents the profit of a company based on the number of items x produced, the domain tells us the feasible range of production. If the domain is ( 6 , ∞ ) , it means the company must produce more than 6 items to make a profit, otherwise the profit function is not defined or yields a non-real result. Similarly, in physics, if f ( x ) represents the velocity of an object, the domain specifies the time interval for which the velocity is valid.
