If sin (arcsin x) `= (sqrt(2))/(4)`, then what is the value of x ?

Options

A. `(sqrt(2))/(4)`
B. `(sqrt(7))/(7)`
C. `(sqrt(2))/(2)`
D. `(sqrt(14))/(4)`

Answer

A

Explanation

Before solving the question, be sure you understand the difference between the meanings of f and f(x). The letter f can refer to a function. Other letters, like g, u, and t can also refer to functions. Then f(x), which is read ''f of x,'' means the value of the function f at x.
The idea in the solution to this question is that the function of the inverse function of x is x. If `f^(-1)` is the inverse function of the function f, then `f(f^(-1)(x))=x`. Here you are working with the sine function and the inverse sine function. If you say that f(x) = sin x, then `f^(-1)(x)=` arcsin x. So sin (arcsin x) = x. If sin (arcsin x) `= (sqrt(2))/(4)`, then since sin (arcsin x) = x, `x = (sqrt(2))/(4)` , (A) is correct.
Let's briefly review the definition of the inverse sine function. The definition of the inverse sine function, which is called arcsin x, is this. If `-1 le x le 1`, then arcsin x is that value y such that `- (pi)/(2)le y le(pi)/(2)` and sin y = x. In order for g to be a function, to each x of the domain of g, g must specify exactly one value. That is, for each specified x, g(x) is not ambiguous. Now when x increases from `- (pi)/(2)` to `(pi)/(2)`, sin x increases from `-1` to 1, so the inverse sine function arcsin x assigns exactly one value y such that `-(pi)/(2) le y le (pi)/(2)` to each x such that `- 1 le x le 1`.