Question

Let g be the function given by g(x) = √(1 - sin^2x).

a) g(x) = cos x
b) g(x) = |cos x|
c) g(x) = sin x
d) None of the above

Answer 1

The function g(x) simplifies to the absolute value of cosine, as the square root of cosine squared equals the absolute value of cosine, thus the answer is g(x) = |cos x|. Option B is correct.

Step-by-step explanation:

The student is asking about the function g(x) = √(1 - sin^2x), which simplifies to the function for cosine or its absolute value due to the Pythagorean trigonometric identity sin^2x + cos^2x = 1.

To find whether g(x) equals cos x, or |cos x|, or sin x, we recognize that the expression under the square root, 1 - sin^2x, simplifies to cos^2x given the aforementioned identity. Therefore, g(x) = √(cos^2x) which equals |cos x| because the square root of a squared value is the absolute value of the original number, encompassing all x in the domain of g(x).

The function g(x) is defined as g(x) = √(1 - sin^2x). To determine which option correctly represents the function, we can simplify g(x). Since sin^2x is a trigonometric identity which is equal to 1 - cos^2x, we can substitute it into the function:

g(x) = √(1 - (1 - cos^2x))

= √(cos^2x)

= |cos(x)|

Therefore, the correct option is b) g(x) = |cos x|.