Limits and Continuity

Lesson 2

Objective

Use interval and function notation to describe the behavior of piecewise functions.

Criteria for Success

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  1. Connect interval notation to inequality notation to identify a domain.
  2. Describe where a function is increasing, decreasing, or constant using interval notation.
  3. Evaluate piecewise functions presented algebraically or graphically, including at transitions between domains. 

Anchor Problems

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Problem 1

Use the graph below to find the value of $${f(-3)}$$.

Guiding Questions

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Problem 2

Below is a new kind of notation that we will use to describe an interval (called interval notation).

In each column, the inequalities and interval notation shown describe the same interval.

Interval notation: $${(2,4)}$$ Interval notation: $${[-1,3]}$$
Inequality notation: $${2<x<4}$$ Inequality notation: $${-1\leq x\leq-3 }$$
Verbal: Over the interval from two to four, exclusive. Verbal: Over the interval from negative one to three, inclusive.

Write the following inequality using interval notation.

$${-2 < x\leq 6}$$

Guiding Questions

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Problem 3

Using interval notation:

  1. On what intervals is the function increasing?
  2. On what intervals is the function descreasing?
  3. On what intervals is the function constant?

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Create Desmos graphs to provide additional practice
  • Include problems where students must find infinite intervals, like writing that $${x>3}$$ is the same as $${(3, \infty)}$$
  • Include problems generating graphs based on information given in function and interval notation

Target Task

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Draw a piecewise function that matches the following constraints.

$${f(4)=6}$$

$${f(6)=8}$$

$${f(8)=12}$$

Increasing function over the interval $${(4,8)}$$