Piecewise Functions and Relations

While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes.  The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:

Piecewise1
Acceleration up to a speed limit
Piecewise2
Free fall then controlled descent

In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.

In the graph on the right, note that the blue curve starts at x = 0.  It does not appear of the left of the vertical axis at all.  And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.

These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function.  Both of these graphs are Continue reading Piecewise Functions and Relations

Domain, Range, and Co-domain of a Function

When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.

Domain

The “domain” of a function or relation is:

  • the set of all values for which it can be evaluated
  • the set of  allowable “input” values
  • the values along the horizontal axis for which a point can be plotted along the vertical axis

For example, the following functions can be evaluated for any value of  “x”:

f(x)=2x+1\\*~\\*g(x)=x^2+5

therefore their domains will be “the set of all real numbers”.

The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:

h(x)=\dfrac{1}{x}~~~~~~~~~\text{x cannot be zero}\\*~\\*j(x)=\dfrac{1}{(x-2)(x+4)}~~~~~~\text{x cannot be 2 or -4}\\*~\\*k(x)=3x+2,~1<x<10~~~~\text{only values between -1 and 10 may be used for x}

The values for which a function or relation cannot be Continue reading Domain, Range, and Co-domain of a Function

Function Translations: How to recognize and analyze them

A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. A function can be translated either vertically, horizontally, or both. Other possible “transformations” of a function include dilation, reflection, and rotation.

Imagine a graph drawn on tracing paper or a transparency, then placed over a separate set of axes. If you move the graph left or right in the direction of the horizontal axis, without rotating it, you are “translating” the graph horizontally. Move the graph straight up or down in the direction of the vertical axis, and you are translating the graph vertically.

In the text that follows, we will explore how we know that the graph of a function like

g(x)=x^2-6x+10\\*~\\*~~~~~~~=(x-3)^2+1

can be described as a translation of the graph of

f(x)=x^2 ~~~~\text{translated right by 3, and up by 1}

Describing g(x) as a translation of a simpler-looking (and more familiar) function like f(x) makes it easier to understand and predict its behavior, and  Continue reading Function Translations: How to recognize and analyze them

Function Notation

Many students seem puzzled by function notation well after it has been introduced, and ask “why can’t we just write y = 3x instead of f(x) = 3x ?”. To motivate the use of function notation and improve understanding, I advocate using multi-variable functions instead of single-variable functions in introducing this notation. My introduction usually proceeds something like this:

A Problem

Suppose you are talking to a friend over the telephone, and they have a piece of paper in front of them on which they have drawn a circle using a ruler and a compass. You need the clearest, most concise instructions possible so that you can exactly duplicate the appearance of the paper your friend has… your circle must be exactly the same size as theirs, and in the exact same location on the piece of paper. What information do you need in order to be able to do this?

For the size of the circle, either the radius or the diameter will work nicely (most students come up with this answer quickly). If you are using a compass to draw the circle, the radius will probably be more convenient. However, the radius alone does not tell you where to locate the circle on the paper. What aspect of a circle will do the best job of Continue reading Function Notation