http://www.ck12.org Chapter 1. Functions and Graphs
Example C
Which function families are unbounded above and below?
Solution:
Look for the function families that don’t have an overall maximum or minimum value.
y=x,y=x^3 ,y=^1 x,y=lnx
Concept Problem Revisited
Whilex^2 and 2xhave similar ingredients, they have very different graphical features. The squaring function is
symmetric about the linex=0 while the exponential function is not. Whenx=0, the squaring function has a height
of zero and the exponential function has a height of one. The squaring function has a slope that becomes steeper as
xgets further from the origin while the exponential function flattens asxgets very small. All of these differences are
important and not obvious at first glance.
Vocabulary
Afunction familyis a group of functions that all have the same basic shape.
Aparameteris a constant embedded in a function that affects the shape of the function in a limited and specific
way.
Unbounded abovemeans that the function gets bigger than any specific number you can choose.
Unbounded belowmeans that the function can get smaller than any specific number you can choose.
Continuousmeans that the function can be drawn entirely without lifting your pencil.
Guided Practice
- Which functions are discontinuous?
- Which functions always have a positive slope over the entire real line?
- Which functions are defined for allxvalues?
Answers: - The reciprocal function is the only function included here that is discontinuous.
2.y=x,y=ex,y= 1 +^1 e−x. Some functions that are close but not quite:y=x^3 ,y=√x
3.y=x,y=x^2 ,y=x^3 ,y=ex,y=sinx,y=|x|,y= 1 +^1 e−x
Practice
For 1-10, sketch a graph of the function from memory.
1.y=ex
2.y=ln(x)
3.y=sin(x)
4.y=x^2
5.y=|x|
6.y=^1 x