# Inverse Functions

Definition. A function is called one-to-one if whenever

Definition. For any function , the inverse of , denoted by , is the set of all pairs for which is in

Theorem. is a function if and only if is one-to-one.

Proof. (1) Assume is one-to-one. Let , be two pairs in . Then and are in . This means . Since is one-to-one, it follows that . Hence is a function.

(2) Assume that is a function. If , then contains the pair and . Therefore and are in . Since is a function, this implies . Hence is one-to-one.

Theorem. If is continuous and one-to-one on an interval, then is either increasing or decreasing on that interval.

Proof. If is neither increasing nor decreasing, then there exist and on the interval with and . Consider the function defined by

Then is continuous on with and . It follows from the Intermediate Value Theorem that there exists such that . This implies since is one-to-one. But the inequality

Theorem. If is continuous and one-to-one on an interval, then is also continuous.

Proof. From the preceding theorem, we see that is either increasing or decreasing on that interval. By replacing , we may assume that is increasing. Fix in the domain of and let . Such a number must be of the form for some in the domain of . We wish to find so that the inequality

implies the inequality

Now let be the smaller of and . This choice of ensures that

Consequently, the inequality

implies the inequality

Since is also increasing, we have

i.e.,

Theorem. If is a continuous one-to-one function defined on an interval and then is not differentiable at .

Proof. Since , the differentiability of at would imply

Theorem. Let a continuous one-to-one function defined on an interval, and suppose that is differentiable at , with derivative . Then is differentiable at , and

Proof. Let . Every number in the domain of can be written in the form

for some unique . From this we have

or

The previous theorem shows that is continuous at . Therefore

i.e.,

Since

it follows from

that

# Trigonometric Functions

Definition.

Definition. For , define

It follows from the Fundamental Theorem of Calculus that is differentiable on and

Therefore decreases from to

Definition. If , then is the unique number in such that

Definition.

Theorem. If , then

Proof. Let . Then . Since is differentiable with , it follows that is differentiable with . Hence

Therefore . Since we have

For , define

For any real number , write

where and an integer so that . Define

Then

Other standard trigonometric functions are defined as:

Theorem. If then

If , then

# Inverse Trigonometric Functions

The inverse of the function

is denoted by or arcsin. Thus is the unique number in satisfying

The domain of is . The inverse of

is denoted by or arccos. The domain of is The is the unique number in satisfying

The inverse of the function

is denoted by or arctan. The domain of is . The is the unique number in satisfying

Theorem. For , we have

For , we have

Theorem. The only solution to the differential equation

is

Proof. The given condition implies

Therefore

so is constant. From it follows that this constant is zero. Thus

This implies

Theorem. If has second derivative everywhere and

then

Proof. Let

Then

Consequently,

Therefore

Theorem.

Proof. Fix . Define . Then

Therefore

The previous theorem shows

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