Relations and Functions
You learned that the most general type of link between two quantities is called a relation. In a relation each x value (domain value) maps to one or more y value (range value)
Example: y2 = x, In this case there are two values of y for every value of x since y is equal to plus or minus the squareroot of x. The requirement for a relation is that there be one or more values of y for every value of x. That requirement is fulfilled by the example.
In a function, as opposed to a relation each x value (domain value) maps to exactly one, and only one range value (y value). Note that all of the x values may map to the same value of y. The requirement is that each x maps to one y value. There is no requiement that each x map to a different value of y. y = 6x + 3 and y = x2 are examples of functions, each x maps to one and only one value of y.
In this session we are going to take a deeper look at functions and the notation that goes with them.
Functional Notation - Input and Output - An example of functional notation is d = g(t) = 16(t)2. In this case we are saying that d (distance traveled) in a function of t, here g(t) and the particular function is 16(t)2. If we want to evaluate distance traveled at a particular time, say 2 seconds, we could write d = g(t=2) = 16(2)2. That is the way we could write it, but we don't write it that way (we use a notational form that was first used by Leonhard Euler around 1734). The notation that is used expresses the idea by writting it as g(2) = 16(2)2. In other words the argument of g( ? ) is the value of t that we want to use and everwhere in the function we replace t with that value. So if we had h(x) = 2x2 + 6x and we wanted to determine the value of the function when x was equal to -4 we would write it as h(-4) = 2(-4)2 + 6(-4) = 2(16) - 24 = 32 - 24 = 8.
To illustrate the idea consider the following examples:
- m(t) = 2(t)3 - 6t2, when t = -1 we have m(-1)=2(-1)3 - 6(-1)2=-2 + 6 = 4
- w(s)= 2s2 - 4s, when s = a we have w(a) = 2(a)2 - 4(a) = 2a2 - 4a
- m(x) = 1/x + 3x, when x = 2x we have m(2x) = 1/(2x) + 3(2x) = 1/2x + 6x
- r(x) = 2x2 - 7x, when x = a- 2 we have
- r(a-2) = 2(a-2)2 - 7(a-2)
- r(a-2 )= 2(a2-4a +4) -7a + 14
- r(a-2)= 2a2-8a + 8- 7a+ 14
- r(a-2)= 2a2 -15a + 22.
Input and Output - When write a function, f(x) the x represents the input, when we have a formula and replace all the x's with the value of x the resulting output is the value of the function for that value of x. Let's look at input and output for three cases, (i) we have a formula, (ii) the information is in a table and (iii) the function is represented by a graph.
Define a function such as: f(x) = 3x2 + 5x - 7
- Select an input value for x, in this case x = -4
- Rewrite the function as: f(-4) = 3(-4)2 + 5(-4) - 7
- Carry out the arithmetic to get: f(-4) = 3(16) - 20 - 7 = 48-27 = 21
- When x = -4 (input) the value of the function is 21 (output)
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In the figure on the right two points have been selected (-4, -1) and (3, 2). We will use the first point to illustrate the treatment of input. f(-4) means the value of the function when x = -4. We go along the x axis to minus 4 and then drop a vertical line to the curve. Next we draw a horizontal line from the curve to the y axis. It itersects the axis at -1 so the value of the function when x = -4 [ f(-4) ] is -1 or f(-4) = -1. Now using the right most point if we are told that f(x) = 2 we go up the y axis (f(x) axis) until we come to 2, then horizontally until we hit the graph and finally vertically until we hit the x axis at x = 3. This says that when f(x) = 2 the value of x is 3. [Remember there could be more than one value of x that gives us this value of f(x)].
Domain of a function: The domain on a function consists of all of the values that the independent variable (usually X ) for which the function is defined.
- f(x) = 1/x is defined for all values of x except x = 0 (you cannot divide by zero)
- f(x) = x2 is defined for all values of x.
Range of a function:The range of a function consists of all of the values of the function as the independent variable (x) varies over its domain.
Using the previous examples
- the range of f(x) = 1/x is from -infinity up to zero and from zero to plus infinity (does not include zero)
- the range of f(x) = x2 is from zero to plus infinity (the function is never negative)