CHAPTER 2 LESSON 6 Special Functions

Vocabulary Step Function- A function whose graph is a series of line segments Greatest Integer Function- A step function, written as f(x)= [[ x ]], where f(x) is the greatest integer less than or equal to x Constant Function- A function, written as f(x)=b, where m=0 so f(x)=b for all x values Identity Function- A function, written as f(x)=x, where m=1 so y=x for all x values Absolute Value Function- A function, written as f(x)= ∣x∣, where f(x)= the positive form of x Piecewise Function- A function that is written using two or more expressions

Greatest Integer What is the greatest integer less than or equal to the following numbers?

Greatest Integer Function

Domain is all real numbers Range is only integers, or Z

Step Function

Domain is all real numbers Range depends on equation

Changes that can be made to step functions If adding a number (inside or outside) to the step function, the graph moves up. If subtracting a number (inside or outside) to the step function, the graph moves down.

Examples

Changes that can be made to step functions If multiplying a number inside the step function, the length of the steps is smaller, but the distance between the steps stays the same. If multiplying a number outside the step function, the length of the steps is the same, but the distance between the steps is increased.

Examples

Changes that can be made to step functions If dividing by a number inside the step function, the length of the step is larger, but the distance between the steps stays the same. If dividing by a number outside the step function, the length of the step is the same, but the distance between the steps is smaller.

Examples

Changes that can be made to step functions If multiplying or dividing by a negative number, the steps go downward instead of upwards, but the rules for step length and distance between steps still apply

Examples

Constant Function

Domain is all real numbers Range is b from f(x)=b

Examples

Identity Function

Identity Function y=x Domain is all real numbers Range is all real numbers

Absolute Value Function

Domain is all real numbers Range depends on equation

Changes that can be made to Absolute Value functions If adding a number inside the absolute value, the graph moves to the left. If adding a number outside the absolute value, the graph moves up.

Examples

Changes that can be made to Absolute Value functions If subtracting a number inside the absolute value, the graph moves to the right. If subtracting a number outside the absolute value, the graph moves down.

Examples

Changes that can be made to Absolute Value functions If multiplying by a number (inside or outside) to the absolute value, the graph is skinnier. If dividing by a number (inside or outside) to the absolute value, the graph is wider.

Example

Changes that can be made to Absolute Value functions If multiplying or dividing by a positive number, the absolute value looks like V. If multiplying or dividing by a negative number, the absolute value looks like Λ.

Example

Piecewise Function

Domain and Range vary depending on the different pieces

Examples

Homework Worksheet 2-6