Question

A variety of trigonometric functions are shown in the answer choices below. Which trigonometric function has an inverse over the domain x/2≤x≤3x/2 A f(x)=cos(x−12)+32 B f(x)=cos(x+π2) C f(x)=sin(x−12)+32 D f(x)=sin(x+π2)

A. f(x)=cos(x− 1/2 )+ 3/2
B. f(x)=cos(x+ π/2 )
C. f(x)=sin(x− 1/2 )+ 3​/2
D. f(x)=sin(x+ π/2)

Answer 1

The trigonometric function with an inverse over the domain x/2≤x≤3x/2 is Option B, f(x)=cos(x+ π/2), because it is equivalent to the inverse sine function after a phase shift.

Step-by-step explanation:

The question asks to identify which trigonometric function has an inverse over the domain x/2≤x≤3x/2. Inverse trigonometric functions only exist for sine, cosine, and tangent functions over specific domains such that they can pass the horizontal line test (thus having an inverse which would pass the vertical line test).

Looking at the options provided, both sine and cosine functions can have inverses but their domains need to be restricted. The usual domains for the inverse sine function (arcsin) and inverse cosine function (arccos) are [-π/2, π/2] and [0, π], respectively. Analyzing the functions provided, f(x)=cos(x+ π/2) is equivalent to the sine function with a phase shift, and since the sine function has an inverse on [-π/2, π/2], it satisfies the conditions stated in the problem. Therefore, the correct option is B. f(x)=cos(x+ π/2).