Functions & Graphs

The Parthenon,Centennial Park, Nashville, Tennessee

Athena Statue (holding Nike),Centennial Park, Nashville, Tennessee

We will start our study of calculus by spending a few days in a quick review of several pre-calculus topics.

Whether you are comfortable with all of these pre-calculus topics or have forgotten many of them and have to learn them now, it is okay either way.

The most important thing to remember in your study of calculus is this:

Don’t panic!

Calculus is a challenging class for everyone.

If you ask questions, keep up with homework, and study for quizzes and tests, this will be the best math class you have

ever taken, and you will be proud of your success.and …

College will be way more fun if you have taken calculus in high school and don’t have to stress over passing your first college math class!

Domain & Range

Functions have an independent variable (often x) and a dependent variable (often y).

In this discussion we are going to use x for the independent variable and y for the dependent variable, but we could use other letters.

domain: The set of all possible x values for a function.

range: The set of all possible y values for a function.

We can specify the domain of a function, or we can look at the nature of the function to determine the natural domain of the function.

Example:

Find the domain and range of: 1f xx

When finding the domain, it is often easier to think of what values that we cannot use for x.

In this case, x cannot equal zero. The domain is:

,0 0,D or 0D x x

We usually use interval notation.

These are open intervals (with parentheses) because we don’t count the zero. Also notice the symbol for union.

This is set builder notation.

1f xx

Looking at the graph, we see that we can have every y value except zero, so the range is:

,0 0,D or 0D x x

x

y

x

y

,0 0,R

Another example:

Find the domain and range of: y x

Since we cannot take the square root of a negative number, the domain is:

This is a half-open interval.

We use a square bracket on the left boundary point because we count the zero. (The interval is closed at zero.)

When infinity is a boundary, it is always an open interval on that end, with parentheses.

[0, )D

x

y

Looking at the graph, we see that there are no negative y values, so the range is:

0,R

Another example:

Find the domain and range of: y x

Since we cannot take the square root of a negative number, the domain is: [0, )D

x

y

x

y

x

y

SymmetryWhen we graph the function , we see that it hasy-axis symmetry.

2y x

When we square the x, any negative sign cancels out, so changing the sign of x does not change the y value.

Any polynomial function with only even exponents behaves the same way, and has y-axis symmetry.

4 22 1y x x

This is , which has an even exponent. 0x

x

y

x

y

Any function with y-axis symmetry is called an even function.

For example, is an even function.

y x

So is . cosy x

x

y

x

y

x

y

A polynomial function with only odd exponents hasorigin symmetry.

Changing the sign of x changes the sign of y.

312

y x

In other words, if (x,y) is on the graph, so is (-x,-y).

2,4

2, 4

x

y

Any function with origin symmetry is called an odd function.

For example, is an odd function.

siny x

x

y

Polynomial functions with exponents that are both even and odd have no symmetry.

x

y

Of course, a graph with x-axis symmetry is not a function at all!

x

y

x y

Fails the vertical line test!

x

y

Piecewise FunctionsWhile many functions can be defined by a single formula, others are defined by applying different formulas to different parts of their domains.

Just graph each piece separately, for each part of the domain.

Example:

2

, 0, 0 1

1, 1

x xy f x x x

x

x

y

Write a piecewise function for the graph at right:

Example:

There are two pieces to this graph, so we need two equations.

1,1 2,1

, 0 11, 1 2

x xf x

x x

The equation for the left hand piece is: y x

For the right hand piece, the equation is: 0 1 1y x

This simplifies to: 1y x

Paying attention to the open and closed circles, we get the piecewise function:

p