Honors Pre-Calculus A, 3rd Hour, Winter 2016: 1.1- Functions

Section 1.1 from Chapter 1 establishes the basic foundation of the chapter to prepare us for the future sections. Section 1.1 focuses mainly on functions.

Definition of a Function:

A relation that assigns to each input exactly one output.

An output is another name for y and can also be described as range (set of outputs). Likewise, input is the x also known as the domain (set of inputs). Therefore, in order for something to be considered a function, there has to be exactly one x for a single y or else it's not a function.

Functions can be represented in the four most common ways:

Verbally- a description of words that explain how the input variable is related to the output variable.

Example: The input value are the sides of a square and the output value is the perimeter

Numerically- the use of tables or ordered pairs to represent the relationship between the inputs and outputs

Example: (0, 1), (1, 2), (2, 3), (3, 4)

Graphically- points on a graph in which the input values are represented on the x-axis and the output values are on the y-axis.

This graph represents a function because no input has more than one output value. How can you determine if a graph is a function by just looking at it? What you can do is what's called the Vertical Line Test. You draw a vertical line anywhere on the graph and if the line hits the graph in more than one spot, then it's NOT a function. Therefore, if the vertical line only hits the graph in one spot, it is a function.

Algebraically- by an equation in two variables, most commonly x and y.

Example:

For each value of x, it has exactly one value of y. Therefore, y is a function of x.

Now that we are able to recognize the different ways functions can be represented, let's take a look at determining whether something is a function represented algebraically.

Does this equation represent y as a function of x?

No, this is not a function because the

indicates that x has two values of y. Therefore, y is not a function of x.

Another important part of functions is what we call Function Notation.

Definition of Function Notation:

When representing an equation as a function, it's easier to reference it by renaming it. When you know an equation describes y as a function of x, you can give the function the name of "f".

So if y=1-x, then we can rewrite that in function notation to become f(x)=1-x. The symbol f(x) is read like the value of f at x. Note that the name of the function is f and f(x) is the output value and the input value is x.

Evaluating a function uses function notation to find values of y when given certain x values. Functions have function values that can be shown as f(-1), f(0) and so on. To find these values you have to substitute the input values into the equations.

Example:

Let f(x)=2x-3 and find f(-1)