# Points, Vectors, and Functions

### Topics

## Introduction to Points, Vectors, And Functions - At A Glance:

What does it mean for something to be increasing or decreasing? Albert is an industrious fellow. He decides to start his own lawn mowing business. When Albert gets paid for mowing his neighbor's lawn, he increases the amount of cash in his wallet. And when Albert decides go buy a pack of cotton candy bubble gum as a reward for his labors, he decrease that cash cache while increasing the tasty goodness in his mouth.

We say a function *y* = *f*(*x*) is **strictly increasing** if *y* gets bigger as *x* moves to the right. A strictly increasing function could look like one of these (or like many other things):

In symbols, if a < b then *f*(*a*) < *f*(*b*).

Sometimes, functions in real life are more complicated. We only deal with these functions in pieces called intervals.

We say a function *y* = *f*(*x*) is strictly increasing *on the interval* *I* if *a* < *b* implies *f*(*a*) < *f*(*b*) for all a, b in the interval *I*.

### Sample Problem

The function *y* = *f*(*x*) graphed below is increasing on the interval [*x*_{1,}*x*_{2}], but not on the whole real line:

We say a function *y* = *f*(*x*) is **nondecreasing** if *y* doesn't get smaller as *x* moves to the right. The function doesn't have to get bigger, but it's not allowed to get smaller.

In symbols, if *a* < *b* then *f*(*a*) ≤ *f*(*b*).

A function can also be nondecreasing on a particular interval.

We can say a function is *increasing at a point* if its slope at that point is positive.

This is different than *increasing over an interval*.

We recommend not using the word *increasing* all by itself. We instead say *strictly increasing* or *nondecreasing* to be clear.

All these definitions were for functions that, more-or-less, increase. There are similar definitions for functions that, more-or-less, go down.

A function *y* = *f*(*x*) is **strictly decreasing** if the function is always getting smaller.

In symbols, if *a* < *b* then *f*(*a*) > *f*(*b*).

A function *y* = *f*(*x*) is **nonincreasing** if the function never gets bigger. The function might stay the same for a bit, but it never increase.

In symbols, if *a* < *b* then *f*(*a*) ≥ *f*(*b*).

**Be Careful:** *strictly increasing* and *strictly decreasing* aren't opposites. It's possible to have a function that's neither:

Similarly, *nondecreasing* and *nonincreasing* aren't opposites. It's possible to have a function that's neither:

There's one other term used when talking about the increasing-ness or decreasing-ness of functions: **monotonic**. No, we aren't talking about Ben Stein. A function is monotonic if it's going only one way. Therefore, it is either going up or going down.

#### Exercise 1

Exercises. Determine if each statement is true or false for a function *f*(*x*) defined on the whole real line.

- If
*f*(*x*) is strictly increasing then*f*(*x*) must also be non-decreasing.

#### Exercise 2

Exercises. Determine if each statement is true or false for a function *f*(*x*) defined on the whole real line.

- If
*f(x)*is non-increasing then*f(x)*must be strictly decreasing.

#### Exercise 3

Exercises. Determine if each statement is true or false for a function *f(x)* defined on the whole real line.

- The function
*f(x)*must be at least one of the following: strictly increasing, non-decreasing, strictly decreasing, or non-increasing.

#### Exercise 4

Exercises. Determine whether each function is (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, (e) none of the above. There may be more than one correct answer.

*y*= 5

#### Exercise 5

Exercises. Determine whether each function is (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, (e) none of the above. There may be more than one correct answer.

*y*=*x*<sup>2</sup>

#### Exercise 6

Exercises. Determine whether each function is (a) strictly increasing, (b) non-decreasing, (c) strictly decreasing, (d) non-increasing, (e) none of the above. There may be more than one correct answer.

*y*=*x*<sup>3</sup>

#### Exercise 7

*y*= -*x*

#### Exercise 8

Consider the function *f*(*x*) graphed below.

What is the largest interval on which *f*(*x*) is

- strictly increasing?

#### Exercise 9

Consider the function *f*(*x*) graphed below.

What is the largest interval on which *f*(*x*) is

- non-decreasing?

#### Exercise 10

Consider the function *f*(*x*) graphed below.

What is the largest interval on which *f*(*x*) is

- strictly decreasing?

#### Exercise 11

Consider the function *f*(*x*) graphed below.

*f*(*x*) is

- non-increasing?