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Characteristics of Functions

There are several characteristics of functions, we’ll look at them below.

 

1. Odd and Even functions

A function can be odd or even. 

We say a function is odd  if they are symmetrical about a point, usually the origin. Such a graph when rotated by 180° about the origin (or the point of symmetry) will give the original graph.

For an odd function, f(-x)  =  – f(x), for all values of x in the domain.

A function is said to be even if it is symmetric about the y-axis. The curve has a line of symmetry about the y-axis and will look like a reflection of itself along the y-axis.

For an even function, f(-x)  =  f(x), for all values of x in the domain.

 

Examples of odd and even functions

To find out if a function f(x) is odd or even, we need to find out what happens when it is turned to f(-x). 

If f(-x) is the same as f(x), i.e. f(-x) = f(x), then it is an even function.

And if f(-x) becomes -f(x), i.e. f(-x) = -f(x), then it is an odd function.

  1. f(x) = x4 + x2 + 2

    f(-x) = (-x)4 + (-x)2 + 2  =  x4 + x2 + 2

    f(-x) is the same as f(x). Hence this is an even function.

    A useful tip to remember is that even functions will all have even powers in the function.

  2.  f(x) = x3 + 3x

    f(-x) = (-x)3 + 3 (-x)

            = -x3 – 3x

            = – (x3 + 3x)

            = -f(x)

    So f(x) = x3 + 3x is an odd function.

    Odd functions will have odd powers in the function.

  3.  f(x) = x3 + x2 + x + 1

    f(-x) = (-x)3 + (-x)2 + (-x) + 1

            = -x3 + x2 -x + 1

    This is neither an even function nor an odd function.

 

2. Increasing and decreasing functions

increasing curve

 

A function is said to be an increasing function when the value of y increases as the values of x increase in the given domain. The curve of an increasing function could be either of the two types shown below. You will notice that the curve moves upwards as we move from left to right.

 

 

 

decreasing curve

A function is said to be a decreasing function when the value of y decreases as the values of x increase in the given domain. The curve of a decreasing function could be either of the two types shown below. You will notice that the curve moves downwards as we move from left to right.

 

 

 

 

 

3. Stationary point

When the curve is neither increasing or decreasing, then we say that the curve is stationary at that point. This point is called the stationary point or turning point.

Around the stationary point, the function f(x) changes its curve direction. For instance, when the function/curve is increasing (going up), and reaches the stationary point, then the curve starts decreasing (goes down). This stationary point is called maximum point of the function in a domain.

Similarly when the function/curve is decreasing (going down), and reaches the stationary point, then the curve starts increasing (goes up). This stationary point is called minimum point of the function in a domain.