Find the domain of the function. [tex]f(x)=\frac{\sqrt{x+7}}{-3 x+8}[/tex]
Write your answer as an interval or union of intervals.
Answer (2)
The domain of the function f ( x ) = − 3 x + 8 x + 7 is given by the intervals [ − 7 , 3 8 ) ∪ ( 3 8 , ∞ ) , where − 7 is included, and 3 8 is excluded. This means that x can take any value from − 7 to just below 3 8 and from just above 3 8 to infinity. Both the square root and the denominator conditions are satisfied within these intervals.
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Solve the inequality x + 7 ≥ 0 to find the values of x for which the square root is defined: x ≥ − 7 .
Solve the equation − 3 x + 8 = 0 to find the values of x for which the denominator is zero: x = 3 8 .
Exclude the values of x that make the denominator zero from the interval obtained in step 1: x = 3 8 .
Express the domain as an interval or union of intervals: [ − 7 , 3 8 ) ∪ ( 3 8 , ∞ ) .
Explanation
Analyze the problem and identify restrictions We are asked to find the domain of the function f ( x ) = − 3 x + 8 x + 7 . 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 two restrictions:
The expression inside the square root must be non-negative, i.e., x + 7 ≥ 0 .
The denominator cannot be zero, i.e., − 3 x + 8 = 0 .
Solve the inequality for the square root First, let's solve the inequality x + 7 ≥ 0 :
x + 7 ≥ 0 x ≥ − 7 So, the expression inside the square root is non-negative when x is greater than or equal to − 7 .
Find the value that makes the denominator zero Next, let's find the value of x that makes the denominator zero: − 3 x + 8 = 0 − 3 x = − 8 x = 3 8 So, the denominator is zero when x = 3 8 . Therefore, x cannot be equal to 3 8 .
Combine the restrictions and write the domain in interval notation Now, we need to combine these two restrictions. We have x ≥ − 7 and x = 3 8 . This means that x can be any value greater than or equal to − 7 , except for 3 8 . In interval notation, this is written as [ − 7 , 3 8 ) ∪ ( 3 8 , ∞ ) .
State the final answer Therefore, the domain of the function f ( x ) = − 3 x + 8 x + 7 is [ − 7 , 3 8 ) ∪ ( 3 8 , ∞ ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if f ( x ) represents the number of items sold at price x , then knowing the domain tells us the range of prices for which the model is valid. If the domain is [ − 7 , 3 8 ) ∪ ( 3 8 , ∞ ) , it means the price x can take any value in this interval. Another example is in physics, where f ( x ) could represent the position of an object at time x . The domain would then represent the time interval for which the position is defined. For instance, time cannot be negative, so the domain would exclude negative values.
