function question

Question

Is the sum of two functions always an even function? Explain. 

Answer 1

a FUNCTION f is even if f(-x) = f(x) & it's odd if f(-x) = -f(x).  A function can be even, odd, or neither. 

 

A polynomial function having all even exponents on the variable is even (note constant term is of degree 0, even)

A polynomial function having all odd exponents on the variable (and no constant term) is odd.

 

This has nothing to do with evaluating functions & examining the resultant values, but if you are given graphs, then you can tell if a function is even, because it will be symmetric w.r.t. y-axis, or odd (will be symmetric w.r.t. Origin).

 

Examples: y = 2x³ ODD

y = x² + 1 EVEN

y = 2x³ + x² +1  NEITHER (with or without the constant term)

 

THE ANSWER TO THE QUESTION IS DEFINITELY NO. I hope you like this explanation better.

 

 

Answer 2

NO

 

A function is even if it has symmetry about the y-axis.

A function is odd if it has symmetry about the origin.

 

Note that these two descriptions do not even begin to include all functions. Pick any random functions you like.

Ex: f(x) = x + 3  and  g(x) = (x+3)²

 

The first is a straight line with a slope of 1 and a y-intercept of 3.

The second is a parabola opening up with a vertex at (-3,0).

 

If you graph them and add the y values at corresponding x values, you get the sum of the functions.

Plot that and you will see that these perfectly ordinary functions do not add up to an even function.

 

x      f(x)      g(x)        ∑

-2       1         1           2

-1       2         4           6

0        3         9           12     y-axis

1        4         16         20

2        5         25         30

 

If it were even, the ∑ values on either side of the y-axis would match. For a more theoretical analysis, add the two functions:

 

f(x) + g(x) = (x+3)² + x + 3 = (x² + 6x + 9) + x + 3

 

∑(x) = x² + 7x + 12

 

This is a parabola which opens up and has its vertex at (-3.5,-.25), and as you know a parabola is symmetric about its vertex, which is clearly not on the y-axis.

 

Hope this helps.