Question

How do you find the inverse of `y=cos x +3` and is it a function?

Answer 1

The inverse function is piecewise and is `x= 2kpi + cos^(-1)(y-3), 2kpi <= x <= 2(k+1)pi, k = 0, +-1, +-2, +-3, ...`. See detailed explanation.

Explanation:

If y = f(x), the inverse is the explicit relation `x = f^(-1)(y)` = a

function of y and, graphically, both give the same graph, in the

same frame.

Here, the inverse is

`x = cos^(-1)( y - 3), 0 <= x <= pi`.

The domain limits `[ 0 - pi ]` for x is attributed to the limits

imposed in the conventional definition of inverse cosine. The graph

for both y = 3 + cos x and its inverse

`x = cos^(-1)( y - 3), 0 <= x <= pi`

See the combined graph, for k = 0.

graph{(y- 3 - cos x)(x-arccos (y-2.99))=0[0 3.14 1.9 4]}

It is wrong to swap (x, y) as (y, x) and write the inverse as

`y = cos^(-1)( x - 3 )`. Here, the two graphs are generally

different, sans particular cases like y = 1 / x.

See the combined graph for the wrong inverse.

graph{(y- 3 - cos x)(y-arccos (x-3))=0[0 3.14 1.9 4]}

For bijectivity ( one for one, either way), I have given piecewise

definition for the inverse. See the graph below that is same for

both.

graph{(y- 3 - cos x)(x-arccos (y-3.01))=0[0 40 1.9 4.1] }