Consider the
f(x)=srt(x+1/x-1) and
g(x)= sqrt(x+1)/sqrt(x-1). Explain why f and g are not the same function. Thank you!
For the community: Please don't submit solutions to this problem. This is a homework problem for an online course that does not allow students to search for answers to specific problems outside of the school.
For the original poster: We realize that homework can be challenging. If you wish to receive help from the staff or other students, we encourage you to use the resources that the online classes provide. Please don't ask or search online for homework help. We understand that it's common in today's information age to look for resources online, and in some contexts, that's a great thing! However, it's against our Honor Code. You can ask for help on the message boards, and you can learn from the official solution after you submit your answer.
To determine whether the functions f(x) = sqrt((x+1)/(x-1)) and g(x) = sqrt(x+1)/sqrt(x-1) are the same, we need to compare their domains and establish if they produce the same output for every input.
1. Domains of f(x) and g(x):
- The function f(x) is defined for any real number x except for x=1, because division by zero is undefined.
- The function g(x) is defined for any real number x such that x > 1, because square roots of negative numbers are undefined in the real number system.
2. Simplifying f(x) and g(x):
- Let's simplify f(x) by multiplying the numerator and the denominator by the conjugate of the denominator, (x+1)/(x+1). This gives us f(x) = sqrt((x+1)/(x-1)) * sqrt((x+1)/(x+1)).
- Simplifying further, f(x) = sqrt((x+1)^2/(x-1)(x+1)) = sqrt((x+1)^2)/(x+1)sqrt(x-1) = (x+1)/sqrt(x-1)(x+1) = (x+1)/sqrt(x-1).
- Simplifying g(x), we have g(x) = sqrt(x+1)/sqrt(x-1).
3. Comparing f(x) and g(x):
- By comparing the simplified forms of f(x) and g(x), we can see that they are not equal.
- f(x) = (x+1)/sqrt(x-1), while g(x) = sqrt(x+1)/sqrt(x-1).
- The main difference is that the square root is applied only to the numerator of g(x).
Therefore, f(x) and g(x) are not the same function because they have different domains and produce different output for certain inputs.