$f(x)=\frac{8}{\sqrt{7-x}}$
Answer (2)
The function is f ( x ) = 7 − x 8 .
The expression inside the square root must be positive: 0"> 7 − x > 0 .
Solving the inequality gives x < 7 .
The domain of the function is ( − ∞ , 7 ) . ( − ∞ , 7 )
Explanation
Understanding the Function We are given the function f ( x ) = 7 − x 8 and we want to analyze it. The main thing to consider is the domain of the function, which is the set of all possible values of x for which the function is defined.
Finding the Domain Condition For the function to be defined, we need the expression inside the square root to be positive, since we cannot take the square root of a negative number. Also, since the square root is in the denominator, it cannot be equal to zero. Therefore, we must have 0"> 7 − x > 0 .
Solving the Inequality Now, we solve the inequality 0"> 7 − x > 0 for x . Adding x to both sides, we get x"> 7 > x , which is equivalent to x < 7 .
Expressing the Domain So, the domain of the function is all real numbers x such that x < 7 . In interval notation, this is ( − ∞ , 7 ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if f ( x ) represents the amount of water remaining in a tank after x minutes, the domain tells us for how long the function is valid. If the domain is ( − ∞ , 7 ) , it means the function only makes sense for the first 7 minutes. After that, the tank is empty, and the function is no longer applicable. This concept is also used in physics, engineering, and economics to define the limits within which a model is valid.
The function f ( x ) = 7 − x 8 is defined for values of x such that x < 7 . Thus, the domain is expressed in interval notation as ( − ∞ , 7 ) . This domain ensures that the expression under the square root is positive, allowing the function to be valid.
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