Determine the domain and range of $y=\frac{3}{x+10}-8$.
Answer (1)
The domain is all real numbers except where the denominator is zero: x + 10 = 0 , so x = − 10 . The domain is ( − ∞ , − 10 ) ∪ ( − 10 , ∞ ) .
The range is all real numbers except where x + 10 3 is zero. Since x + 10 3 can be any value except 0, y can be any value except − 8 .
The range is ( − ∞ , − 8 ) ∪ ( − 8 , ∞ ) .
Therefore, the domain is ( − ∞ , − 10 ) ∪ ( − 10 , ∞ ) and the range is ( − ∞ , − 8 ) ∪ ( − 8 , ∞ ) . Domain: ( − ∞ , − 10 ) ∪ ( − 10 , ∞ ) , Range: ( − ∞ , − 8 ) ∪ ( − 8 , ∞ )
Explanation
Analyzing the Domain We are given the function y = x + 10 3 − 8 and we need to determine its domain and range. Let's start by analyzing the domain.
Finding Restrictions on x The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is a rational function, which means it is defined for all real numbers except where the denominator is equal to zero. So, we need to find the value(s) of x for which x + 10 = 0 .
Determining the Domain Solving the equation x + 10 = 0 , we get x = − 10 . This means that the function is undefined when x = − 10 . Therefore, the domain of the function is all real numbers except x = − 10 . In interval notation, the domain is ( − ∞ , − 10 ) ∪ ( − 10 , ∞ ) .
Analyzing the Range Now, let's analyze the range. The range of a function is the set of all possible output values (y-values). To find the range, we can consider the behavior of the function as x approaches the excluded value and as x approaches infinity.
Finding Restrictions on y As x approaches − 10 , the term x + 10 3 approaches either positive or negative infinity, depending on whether x approaches − 10 from the left or the right. Therefore, y can take on any value except for the value it approaches as x + 10 3 approaches 0.
Determining the Range The term x + 10 3 can take any value except 0, because the numerator is a non-zero constant. Therefore, y = x + 10 3 − 8 can take any value except − 8 . This is because as x goes to ± ∞ , x + 10 3 approaches 0, and y approaches − 8 . So, the range of the function is all real numbers except y = − 8 . In interval notation, the range is ( − ∞ , − 8 ) ∪ ( − 8 , ∞ ) .
Final Answer In summary, the domain of the function y = x + 10 3 − 8 is ( − ∞ , − 10 ) ∪ ( − 10 , ∞ ) , and the range is ( − ∞ , − 8 ) ∪ ( − 8 , ∞ ) .
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, consider a scenario where you are designing a website. The function y = x + 10 3 − 8 could represent the loading time of a webpage, where x is the number of users accessing the site simultaneously. The domain tells you the number of users the site can handle without crashing (in this case, not -10, which is nonsensical in this context but mathematically important). The range tells you the possible loading times users might experience. Knowing these limits helps you optimize the website's performance and ensure a smooth user experience. By understanding the behavior of such functions, you can make informed decisions about resource allocation and system design.
