Math Is Fun Forum

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

even and composed functions

If g(x) is an even function then f(g(x)) is even for every
function f(x).

solution :

True. We have g(−x) = g(x) since g is even, and therefore f(g(−x)) = f(g(x))

is that mean wherever I compose f(even function input) the output will be even?

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: even and composed functions

is that mean wherever I compose f(even function input) the output will be even?

The actual number that is output may be even or odd.

eg.  g(x) = 1 - x^2                g(3) = 1 - 9 = -8

If f(x) = x + 2 then f(g(3)) = f(-8) =  -8 + 2 = -6

But the composite function will be even.

eg with f and g as above

f(g(x)) = 1 - x^2 + 2 = 3 - x^2

f(g(-x)) = 1 - (-x)^2 + 2 = 3 - x^2 = f(g(x))       so fg is an even function.

This will always be true when g is even whatever f is.

Bob

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pamshaw

Member

Registered: 2021-12-07

Posts: 21

Re: even and composed functions

In math functions that satisfy particular symmetric relations by taking additive inverse are known as even function and odd functions. They are very important in theory of Fourier series and power series. It is also very important in many areas of analysis of mathematics.
Even Functions
A function f will be even if the x and –x holds for all in the domain of f
f(x) = f(-x)
In geometry even function’s graph is symmetric with respect to axis Y. This means after reflecting with the Y-axis no change will happen in the graph.
X2, |x| and cos x are some examples of even functions.
Odd Functions
A function f will be odd if the x and –x holds for all in the domain of f
-f(x) = f (-x)
In geometry, an odd function’s graph has rotational symmetric with respect of origin. This means after rotation of 180° with origin no changing will happen in the graph
x, x³ and sin x are some example of odd function