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

The inverse function is piecewise and is

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

same frame.

Here, the inverse is

imposed in the conventional definition of inverse cosine. The graph

for both y = 3 + cos x and its inverse

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

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] }

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To find the inverse of ( y = \cos(x) + 3 ), first, swap the roles of x and y. Then, solve for y.

[ x = \cos(y) + 3 ]

Now, isolate ( \cos(y) ).

[ \cos(y) = x - 3 ]

To find ( y ), take the inverse cosine of both sides.

[ y = \arccos(x - 3) ]

Whether this inverse is a function depends on the domain and range of the original function. Since the range of ( \cos(x) ) is from -1 to 1, when you add 3 to it, the range of ( y = \cos(x) + 3 ) is from 2 to 4. The domain of the inverse function will be the same as the range of the original function, and the range of the inverse function will be the same as the domain of the original function. Therefore, yes, the inverse is a function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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