Some functions are increasing on one interval and decreasing on another. The function f(x) = x2 shown in the graph below is decreasing on the interval ��,0] but increasing on [0,��. The function f(x) = x2 The function f(x) = x2 is symmetrical about the y axis, the y axis acts as a mirror reflecting the graph. The y axis, ( x = 0 ) is the axis of symmetry of the function. Functions that are symmetrical about the y axis are called even functions. It's usually simple to see from the graph if a function is an even function . We can prove it algebraically by comparing f(a) and f(�a). Example 1 Is an even function?. The calculator shows a graph that seems to be even.  To prove it we calculate f( -a ) This means that f(x) is an even function. Example 2 Is f(x) = 8(ex � x � 1) an even function? The calculator shows a graph that looks a bit like a parabola. Now we'll do the algebra: f(�a) = 8(e�a + a � 1) f(a) = 8(ea � a � 1) These are not the same and therefore the function is not even. A second type of symmetry is symmetry about the origin. A graph that is the same after we have turned it through 180� about the point (0, 0) is symmetrical about the origin. ( you can see this by turning the graph upside down ). A function with this property is called an odd function. An example of an odd function is f(x) = x3. The arrows show that when the graph of f(x) = x3 is turned half a circle about the origin it falls exactly over the original graph. To show that a function is odd we have to show that f(�a) = �f(a). Example 3 Is an odd function? The calculator shows a graph that seems to be symmetrical about the origin. To prove it we compare f( a ) and f( -a ) We see that f(�a) = �f(a) so f(x) is an odd function. If we translate an even function we get a function that is symmetrical about a vertical line. We can see this if we think about the graph of a parabolic function , it always has a vertical axis of symmetry. If we translate an odd function then we get a function that is symmetrical about a point. Example 4 Is the graph of The calculator shows a graph that seems to be symmetrical about a vertical line near to x = �1. We can rewrite the function as follows: The function is obviously a translation of the function So g(x) is an even function and f(x) is found by translating g(x) by one unit to the left. Example 5 Is the function The graph seems to suggest it is symmetrical about a point. The function Now we'll check that g(x) is an odd function. We conclude that g(x) is an odd function and f(x) is found by translating g(x) one unit to the right. So f(x) is symmetrical about the point (1, 0). If we have a function, say f, defined on a set, say R such that the function takes every value in the set R, then the function f is called a surjection. It's more common to say that the function is " onto". If there are values in the set R that the function f does not take then f is not a surjection and the word "into" is commonly used. Example The function y = f(x) = x + 1 defined on R has a straight line graph as the diagram shows. Whatever value of y we choose, we can always find an x that maps onto the y we have chosen. So the function f(x) defined on R is a surjection. We say that it is a mapping from R onto R.
An example of a function that is not "onto" R is f(x) = x2. The table of values show that no negative values appear in the y column.
there can be no negative numbers in this column If, however we limit ourselves to positive real numbers R+ then we can find an x corresponding to every y value in R+ so g(x) = x2 is a mapping from R+ onto R+ and g(x) is therefore "onto" R+ . The diagram below shows a function f mapping values from the domain A onto the codomain B. If the function f is a surjection ( onto ) then every element in the set B is a mapping of some element from the set A. In other words the range of the function is the same as the chosen codomain. The next diagram shows shows a function g that maps the elements of the domain A onto just a small part of the codomain B. The shaded area of B shows the range of the function . It's only a part of B. In other words there are elements in B that are not used by g and g is therefore not a mapping from A onto B but a mapping from A into B. So g is not a surjection. We are now going to look at another property of functions. This is a property called one - to - one. It means that different x values always map onto different y values. In other words if we make a table of values for the function then the same y value never appears more than once in the table. The formal definition is as follows: A function f is one - to - one If a function is continuous and one - to - one then it is either always increasing or always decreasing. An easy way to see this on a graph is to draw a horizontal line through the graph . If the line only cuts the curve once then the function is one - to - one. This must of course apply to all horizontal lines that can be drawn through the graph. Example 6 Is the function If the function is one - to - one the equation The graph is shown below and it is easily seen that a horizontal line will only cut the curve once. This applies to more than just the interval ��2,�� . Example 7 Are the functions f(x) = x2 and g(x) = x3 defined on R one - to - one? f(�1) = (�1)2 = 1 = f(1). There are two values of x that give the y value 1 so the function is not one - to - one. f(x) is a parabola and a horizontal line can cut it twice. g(x) = x3 is one - to - one because x13 = x23 means that x1 = x2 . The graph is shown below and it can easily be seen that a horizontal line only cuts the curve once Functions that are both one to one and onto are called bijections The function g(x) = x3 in example 7 is both one - to - one and onto. y values go from y = �∞ to y = ∞ and the function is increasing on all it's domain. |
Which table shows a function that is increasing only over the interval (-2 1)
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