Set & Interval Notation, Domain, Range, End Behavior

Author:
Lydia Pernia, Tim Brzezinski
Topic:
Algebra

Finite and Infinite Intervals.

Explore the applet above. Move the bottom slider to see the different types of intervals. Answer the following questions.

Q1.

In you own words what is an interval? (Not the notation)

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Q2

When are ( ) used?

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Q3

When are [ ] used?

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Q4

Based on what you have seen what do you think is the difference between a finite interval and an infinite interval?

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Interval Representations

Intervals can be represented using graphs, inequalities, interval notation or set notation. Below is an example of each. x > 5 [5, + ) {x| x > 5} read "the set of all real numbers such that x is greater than 5. 4 ≤ x < 10 [4, 10) {x| 4 ≤ x < 10}

Q5

Using the explanation above describe the differences in set notation and interval notation. Which do you prefer?

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Q6

Write the set notation for (6, +∞)

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Q7

Write the set notation for (-3, 8).

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Domain and Range

Examine the applet below to determine the domain and range of the finite functions.

Q8

In your own words describe the domain of a function.

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Q9

In your own words describe the range of a function.

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End behavior of functions.

Examine what happens to the function as x goes to +∞ and then as it goes to -∞ in the applet below.

Linear Equation

Change a to 1 and n to 1 (you change the a and n values either by typing in the box or moving the slide). You have created the parent function of a linear equation. y = x or f(x) = x. The line goes through the origin (0,0) and has a slope of 1, a represent the slope. Notice that as x gets bigger or approaches +∞ y also gets bigger and approaches +∞. In symbols As x→+∞, f(x) →+∞. IGNORE THE lim NOTATION FOR NOW.

Q10

Using the same function f(x) = x a=1, n=1 describe in words what happens when x →-∞

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Q11

Change the a to -1 and leave n =1. Your function is f(x) = -x. What happened to the line?

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Q12

What is the end behavior of f(x) = -x?

Select all that apply
  • A
  • B
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Q13 Go back to the applet and change the a back to 1 and n to 2.

The function is f(x) = x². What is this function called?

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  • A
  • B
  • C
  • D
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Q14

Describe the end behavior of f(x)= x². Are both ends going in the same direction? What happens to the y of f(x) as the x gets bigger? What happens to the y as x gets smaller?

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Q15

What is the end behavior of f(x) = x²?

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  • A
  • B
  • C
  • D
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Q16 Change the a to -1 leave n=2 f(x)=-x²

What happened to the parabola?

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Q17

What is the end behavior of f(x) = -x²?

Select all that apply
  • A
  • B
  • C
  • D
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Q18 Change the a to 1 an n to 3. f(x)=x³

Describe the end behavior of f(x)= x³. Are both ends going in the same direction? What happens to the y of f(x) as the x gets bigger? What happens to the y as x gets smaller?

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Q19

What do you think will happen if a = -1 ? Verify your answer using the applet.

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Q20 Change the n=4, n=5, n=7, n=8

Describe the difference in the shapes of the odd and even exponents (n).

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