How do you know if a function is increasing versus decreasing? How do you know if a function is odd or even? Give an example of each type.

Question

How do you know if a function is increasing versus decreasing?

How do you know if a function is odd or even? Give an example of each type.

Answer 1

Increasing is determined by the behavior, over a designated interval, where each successive y value, for associated increasing x value, is larger than the former y value (for a lesser x value).  On the graph, the 'curve' (function) is tending upward and to the right, as successively larger x values are selected.

 

Decreasing means that a function is tending downward & to the right.

 

In calculus, the derivative's value(s) tip us off to where a function is increasing/decreasing.

 

f(x) is EVEN if f(-x) = f(x)

f(x) is ODD if f(-x) = -f(x).

 

For a polynomial function, it's even if all exponents on the variable are even (includes constant term, kx^o), and if all exponents of a polynomial are odd (precludes there being a constant-only term), then the function is odd.

Answer 2

Linette,

A few more common examples of odd/even that you are likely to see, in addition to those provided by the tutors:

cos(x) is even because cos(-x) = cos(x)

sin(x) is odd because sin(-x) = -sin(x)

A few rules:

The product of two even functions is even.

The product of two odd functions is even.

The product of an even function and an odd function is odd.

A function f(x) can be separated into an odd part and an even part such that f(x) = f_odd(x) + f_even(x) with

 

f_even(x) = [f(x) + f(-x)] / 2

 

f_odd(x) = [f(x) - f(-x)] / 2