Which trigonometric function requires a domain restriction of to make it invertable?
f(x)=sin x
f(x)=cos x
f(x)=tan x
f(x)=csc x
f(x)=sec x
f(x)=cot x
f(x)=tan x
f(x)=cos x is correct
explain how
The trigonometric function that requires a domain restriction in order to make it invertible is \( f(x) = \cos(x) \).
For a function to be invertible, each input value must correspond to a unique output value. In the case of the cosine function, it is not invertible over its full range \((- \infty, \infty)\) because the cosine function does not pass the horizontal line test (it fails the vertical line test). This means that for some values of \( x \), there are multiple values of \( \cos(x) \).
To make the cosine function invertible, we restrict its domain to the interval \([0, \pi]\) (or any other interval of length \(2\pi\) where \(\cos(x)\) is one-to-one), where the function is one-to-one and therefore can have an inverse function. This restriction ensures that each input value has a unique output value, allowing for the inverse function \( \cos^{-1}(x) \) to exist.