To determine which function is equivalent to the inverse of [tex]\( y = \sec(x) \)[/tex], we need to find an expression for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] by finding the inverse function of [tex]\(\sec(x)\)[/tex].
1. Recall the definition of the secant function:
[tex]\[
y = \sec(x) = \frac{1}{\cos(x)}
\][/tex]
2. To find the inverse, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
[tex]\[
\sec(x) = y
\][/tex]
3. From the definition of the secant function, we know:
[tex]\[
\sec(x) = \frac{1}{\cos(x)}
\][/tex]
Therefore, we can write:
[tex]\[
y = \frac{1}{\cos(x)}
\][/tex]
4. To isolate [tex]\(\cos(x)\)[/tex], take the reciprocal of both sides:
[tex]\[
\cos(x) = \frac{1}{y}
\][/tex]
5. Taking the arccosine (inverse cosine) of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \cos^{-1}\left(\frac{1}{y}\right)
\][/tex]
Thus, the function that is equivalent to the inverse of [tex]\( y = \sec(x) \)[/tex] is:
[tex]\[
y = \cos^{-1}\left(\frac{1}{x}\right)
\][/tex]
To sum up, the correct option is:
[tex]\[
\boxed{y = \cos^{-1}\left(\frac{1}{x}\right)}
\][/tex]
