Which of the following is INCORRECT...Please help me

Question

Which of the following is INCORRECT about the function (f+g)(x), where f(x)=sqrt (x-1) and g(x) = x+2

A. The range of the function is all real

B. The domain of the function is [1,∞)

C. (f+g)(x) = sqrt (x-1)+x+2

D. (f+g)(1)=3

E. (f+g)(2)=5

Answer 1

Which of the following is INCORRECT about the function (f+g)(x), where f(x)=sqrt (x-1) and g(x) = x+2

A. The range of the function is all real

B. The domain of the function is [1,∞)

C. (f+g)(x) = sqrt (x-1)+x+2

D. (f+g)(1)=3

E. (f+g)(2)=5

First, it is easy to verify D and E as true.

D: (f+g)(x) is the same as f(x) + g(x). So, (f+g)(1) = f(1) + g(1) = sqrt(1-1) + 1 + 2 = 0 + 1 + 2 = 3.

E: Same as above, sqrt(2-1) + 2 + 2 = 1 + 2 + 2 = 5.

C: We verified this in order to solve D and E.

B: This is true as well. The function sqrt(x) is not defined for any real number less than zero. Notice that sqrt(x-1) shifts the function to the right by 1. The linear function will not shift the sqrt(x-1) to the left for any reason as it is linear positive increasing. Thus, this function is only defined for real numbers from [1, infinity)

This leaves us with A. By process of elimination, we know this is the correct answer. Just for clarity, however, we know that the range of this function is not all real numbers. These are two positive increasing functions which never decrease at all over the real numbers. Thus, the range will only ever cover the positive real numbers. Additionally, the lowest value for the domain of this function is at x=1, as we just explained. The value of the function here is 3, as we found in part D. 3 is greater than all the real numbers between 0 and 3, so not only does the range of this function not include all real numbers, it doesnt even include all the positive real numbers as it does not ever take a value between 0 and 3.

I hope these answers are clear and help you out! Good luck!