## Functions

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A *function* can be visualized as a machine that takes in an input and returns an output . The collection of all possible inputs is called the *domain*, and the collection of all possible outputs is called the *range*.

This course deals with functions whose domains and ranges are or subsets of (this is the notation for the real numbers).

## Examples

- Polynomials, e.g. . Give the domain and range of .

The domain is , because we can plug in any real number into a polynomial. The range is , which we see by noting that this is a cubic function, so as , , and as , .

- Trigonometric functions, e.g. , , . Give the domain and range for each of these.

For and : domain is ; range is .

For , the domain is ; range is .

- The exponential function, . Give the domain and range for the exponential.

Domain is ; range is .

- The natural logarithm function, . Recall that this is the inverse of the exponential function. Give the domain and range for .

Domain is ; range is . Notice how the domain and range of the exponential relate to the domain and range of the natural logarithm.

- Is a function? If so, why? If not, is there a way to make it into a function?

## Operations on Functions

### Composition

The *composition* of two functions, and , is defined to be the function that takes as its input x and returns as its output fed into .

**Example**:

can be thought of as the composition of two functions, and . If , would be the function that takes an input and returns its square root.

**Example**:

Compute the composition , i.e. the composition of with itself, where .

We find that

### Inverse

The *inverse* is the function that undoes . If you plug into , you will get . Notice that this function works both ways. If you plug into , you will get back again.

NOTE: denotes the inverse, not the reciprocal. .

**Example**:

Let’s consider . Its inverse is .

Notice that the graphs of and are always going to be symmetric about the line . That is the line where the input and the output are the same:

## Classes of Functions

### Polynomials

A polynomial is a function of the form

The top power is called the degree of the polynomial. We can also write a polynomial using a summation notation.

### Rational functions

Rational functions are functions of the form where each is a polynomial.

**Example**:

is a rational function. You have to be careful of the denominator. When the denominator takes a value of zero, the function may not be well-defined.

### Powers

Power functions are functions of the form , where and are constant real numbers.

Other powers besides those of positive integers are useful.

**Example**:

What is ?

What is ?

Recall a fractional power denotes root. For example, . The negative sign in the exponent means that we take the reciprocal. So, .

What is ?

One can rewrite this as . That means we take to the 22nd power and then take the 7th root of the result.

What is ? We are not yet equipped to handle this, but we will come back to it later.

### Trigonometrics

You should be familiar with the basic trigonometric functions , . One fact to keep in mind is for any . This is known as a *Pythagorean identity*, which is so named because of one of the ways to prove it:

By looking at a right triangle with hypotenuse 1 and angle , and labeling the adjacent and opposite sides accordingly, one finds by using Pythagoras' Theorem that .

Another way to think about it is to embed the above triangle into a diagram for the unit circle where we see that and returns the x and y coordinates, respectively, of a point on the unit circle with angle to the -axis:

That explains the nature of the formula . It comes from the equation of the unit circle .

Others trigonometric functions:

, the reciprocal of

, the reciprocal of the

, the reciprocal of the

All four of these have vertical asymptotes at the points where the denominator goes to zero.

### Inverse Trigonometrics

We often write to denote the inverse, but this can cause confusion. Be careful that . To avoid the confusion, the terminology is recommended for the inverse of the function.

The function takes on values and has a restricted domain .

The function likewise has a restricted domain , but it takes values .

The function has an unbounded domain, it is well defined for all inputs. But it has a restricted range .

### Exponentials

Exponential functions are of the form , where is some positive constant. The most common such function, referred to as *the* exponential, is . This is the most common because of its nice integral and differential properties (below).

Algebraic properties of the exponential function:

Differential/integral properties:

Recall the graph of , plotted here alongside its inverse, :

Note that the graphs are symmetric about the line (as is true of the graphs of a function and its inverse).

Before continuing, one might ask, what is ? There are several ways to define , which will be revealed soon. For now, it is an irrational number which is approximately 2.718281828.

## Euler’s Formula

To close this lesson, we give a wonderful formula, which for now we will just take as a fact:

Euler's Formula

The in the exponent is the imaginary number . It has the properties . is not a real number. That doesn't mean that it doesn't exist. It just means it is not on a real number line.

Euler's formula concerns the exponentiation of an imaginary variable. What exactly does that mean? How is this related to trigonometric functions? This will be covered in our next lesson.

## Additional Examples

**Example**

Find the domain of

Note that the square root is only defined when its input is non-negative. Also, the denominator in a rational function cannot be 0. So we find that this function is well-defined if and only if . Factoring gives

By plotting the points and (where the denominator equals 0) and testing points between them, one finds that when or :

So the domain of is or . In interval notation, this is .

**Example**

Find the domain of

Since is only defined on the positive real numbers, we must have . Factoring gives

As in the above example, plotting the points where this equals 0 and then testing points, we find that the domain is and . In interval notation, this is .

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