Question

Determine whether the function\(g(x) = \cos {\kern 1pt} {\kern 1pt} x\)is one-to-one.

Short answer

The function \(g(x) = \cos {\kern 1pt} {\kern 1pt} x\) is not one-to-one.

Step-by-step solution

Given data

The given function is\(g(x) = \cos {\kern 1pt} {\kern 1pt} x\).

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Simplify the expression

Perform the horizontal line test to check whether the function \(g(x) = \cos {\kern 1pt} {\kern 1pt} x\) is one-to-one.

Sketch the graph of the function \(g(x) = \cos {\kern 1pt} {\kern 1pt} x\) and perform the horizontal line test.

From Figure, it is observed that the horizontal line intersects the graph of \(g(x) = \cos {\kern 1pt} {\kern 1pt} x\)at two points, which means it does not pass the horizontal line test.

Therefore, the function \(g(x) = \cos {\kern 1pt} {\kern 1pt} x\) is not one-to-one.