Which of the following functions has a range of `-1 lt y lt 1` ?

Options

A. y = sin x
B. y = cos x
C. `y = (x)/(1+x)`
D. `y=(x)/(sqrt(1+x^(2)))`

Answer

D

Explanation

You should know what the ranges are of (A) y = sin x and (B) y = cos x. They're both `- 1 le y le 1`. They both include -1 and 1 thermselves, so you can eliminate (A) and (B) . The upper limit of (C ) is 1, but (C ) can be much smaller than -1, such as when `x=-0.99`. So you can eliminate (C ). (D) can never be less than zero because the numerator and denominator are nonnegative for all possible x.
The answer is (D) because as x gets very large, `(x)/(sqrt(1+x^(2)))` approaches `(x)/(sqrt(x^(2)))` or `(x)/(x)=1`, so the value of the whole expression approaches 1, and as x gets very small (that is, very negative), `(x)/(sqrt(1+x^(2)))` approaches `(x)/(sqrt(x^(2)))=(x)/(|x|)=-1`, so the value of the whole expression approaches -1.