**Precalculus Chapter 1.1 Functions and their Graphs**

Definition of a

**Function**: A function “f” from a Set A to a Set B is a relation that assigns to each element “x” in the set A exactly one element “y” in the Set B. The Set A is the domain (or set of inputs) of the function “f”, and the Set B contains the range (or set of outputs).

**Example:**

Set A contains {1, 2, 3, 4, 5} and Set B contains {a, b, c, d, e }

Let's say that Set A maps to Set B in giving the following ordered pairs:

{(1, a), (2, c), (3, c), (4, d), (5, b)}

The mapping would look like:

Since every input there is only one output, therefore this is a function.

Remember, for every input there is only one output.

{(a, 2), (b, 1), (c, 4)}

Where a, b, and c are the input and 1, 2, and 4 are the output

1). Each element in Set A must be matched with an element of Set B.

2). Some elements in Set B may no be matched with any elements of Set A.

3). Two or more elements of Set A may be matched with the same element of Set B.

1).

2).

3).

4).

example: y = x

“x” is the independent variable (input or domain)

“y” is the dependent variable (output or range)

a) 2x + y – 6 = 0 Solve for y.

y = -2x + 6

To each value of “x”, there is only one “y” , therefore this example is a function.

b) 3x

2y = -3x

y = (1/2)(-3x

Again, this is a function.

{(1, a), (2, c), (3, c), (4, d), (5, b)}

The mapping would look like:

Since every input there is only one output, therefore this is a function.

Remember, for every input there is only one output.

{(a, 2), (b, 1), (c, 4)}

Where a, b, and c are the input and 1, 2, and 4 are the output

**CHARACTERISTICS of a FUNCTION:**

1). Each element in Set A must be matched with an element of Set B.

2). Some elements in Set B may no be matched with any elements of Set A.

3). Two or more elements of Set A may be matched with the same element of Set B.

**FUNCTIONS**are commonly represented in four ways:1).

**Verbally**– by a sentence that describes how the input variable is related to the output variable.2).

**Numerically**– by a table or list of ordered pairs3).

**Graphically**4).

**Algebraically**by an equation in two variables.example: y = x

^{2}where “y” is a function of “x”“x” is the independent variable (input or domain)

“y” is the dependent variable (output or range)

**TESTING FOR FUNCTION REPRESENTED**:**1) Algebraically**:a) 2x + y – 6 = 0 Solve for y.

y = -2x + 6

To each value of “x”, there is only one “y” , therefore this example is a function.

b) 3x

^{2}+ 2y = 12y = -3x

^{2}+ 1y = (1/2)(-3x

^{2}+ 1)Again, this is a function.

c) 4

^{2}+ 3y

^{2}= 6

3y

^{2}= -4x

^{2}- 6

y

^{2}= (1/3)( -4x

^{2}- 6)

This is not a function!

**2). Vertical line test**– graph the equation. If you can draw a vertical line and it only touches the graph of the equation once, then the equation is a function. If it touches two or more points, then it is not a function.

**3) Function Notation**:

Input = x

Output was y, it will now be f(x)

**Equation example**:

**Was**: y = 2x + 3

Will now be represented by: f (x) = 2x + 3

**EVALUATING A FUNCTION**:

**Example**: f (x) = 2x – 3

f ( 1) = 2 (1) – 3 = 2 – 3 = -1

f (-3) = 2 (-3) – 3 = -6 – 3 = - 9

f (x – 1) = 2 (x – 1) – 3 = 2x – 2 – 3 = 2x – 5

**Example:**g (y) = 7 – 3y

a. g (0) = 7 – 3(0) = 7

b. g (7/3) = 7 – 3 (7/3) = 7 – 7 = 0

c. g ( s + 2) = 7 – 3 (s + 2 ) = 7 – 3s – 6 = 1 – 3s

**A PIECEWISE-DEFINED FUNCTION**

- this means that the function is defined by two or more equations over a specified domain.

f (x ) =

{ 2x + 1, x

`≤`0{ 2x + 2, x is greater than 0

Evaluate: f ( -1)

because -1 is less than 0, we use 2x + 1 so

f (-1) = 2 (-1) + 1 = -2 + 1 = -1

Evaluate: f (0 ) because 0 is equal to 0, we use 2x + 2 so

f (0) = 2x + 2 = 2 (0) + 2 = 2 f (2) = 2x + 2 = 2 (2) + 2 = 4 + 2 = 6