Does f(x)= 1 + cos(x), have an inverse or not, if so, what is the inverse? please explain.
To determine if the function f(x) = 1 + cos(x) has an inverse, let's first consider its domain and range.
The function f(x) = 1 + cos(x) is defined for all real numbers, as the cosine function has a range of [-1, 1], and adding 1 gives a range of [0, 2].
To check for the existence of an inverse, we need to determine if f(x) is a one-to-one (or injective) function. For a function to have an inverse, every value in its domain must correspond to a unique value in its range.
Let's examine the graph of f(x) = 1 + cos(x):
^
2 | *
| /
| /
| /
1 | *
| *
|
0 |_______________________
0 π/2 π 3π/2 2π
From the graph, we can see that f(x) is not a one-to-one function. Specifically, over one period of the cosine function (from 0 to 2π), there are multiple x-values that map to the same y-value. This means that f(x) fails the horizontal line test and does not have an inverse.
Therefore, f(x) = 1 + cos(x) does not have an inverse.