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Step Function

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Step Function

Study Guide

Key Definition

A step function, also known as a staircase function, is a piecewise-defined function that appears as 'steps' on a graph. The basic form is $f(x) = \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Important Notes

The step function $f(x) = \lfloor x \rfloor$ rounds down to the nearest integer.
For negative numbers, the greatest integer is more negative.
Each step in the function represents a constant value over an interval.
Step functions are used in real-world scenarios like billing.
The graph of a step function consists of horizontal segments.

Mathematical Notation

$\lfloor x \rfloor$ represents the greatest integer function

$\lfloor 3.2 \rfloor$ equals 3

$\lfloor -1.3 \rfloor$ equals -2

Remember to use proper notation when solving problems

Why It Works

The step function simplifies continuous functions into discrete steps, making calculations easier in certain contexts like digital signals.

Remember

For any real number $x$, $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1$

Quick Reference

Basic Form:$f(x) = \lfloor x \rfloor$

Example:$\lfloor 5.7 \rfloor = 5$

Understanding Step Function

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A step function is like a staircase on a graph with $f(x) = \lfloor x \rfloor$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\lfloor 4.7 \rfloor$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have $20 to spend on games that cost $3.75 each. How many games can you buy with your money?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find $\lfloor -2.3 \rfloor$ and explain why it is not $-2$.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is $\lfloor \pi \rfloor$?

Recap

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Review key concepts and takeaways