# composite functions

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composite functions

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If you take the output from one function: say, f(x)=sin(x) and use it as the input to another function: say, g(x)= |x| you get a composite function. This would be written as g[f(x)]= |f(x)| but could be simplified to: g(x)=|sin(x)|

The first function makes a sine curve that alternates between positive and negative, with max and min at +1/-1.

The second function says “any negatives become positive”.

Together, you get a function that looks like a sine curve with all negative values reflected to positive along the x axis.

The example I gave is a bit silly, but sometimes composite functions are really handy.

Think of a function as something that alters the input.

For example altering your straight shoe lace by making a loop. That would be loop(lace).

An inverse function as the inverse process like unloop (loopedlace).

Compositive functions as two or more processes one after the other.
Loop(lace(shoeputon(sockputon(gotupinthemorning)))

A function takes an input and gives you a consistent output.

Example:
*f*(x) = x^2
*f*(3) = (3)^2 = 9

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A composite function can be thought of as chained functions. You give it an input and the first function gives you an output, that output is then used as the input for the second function.

Example:
*f*(x) = x^2 , *g*(x) = 4x

*g*(*f*(x)) or (g∘f)(x) means to do *f*(x) first then *g*(x)

*g*(*f*(3)) = *g*(3^2 ) = *g*(9) = 4•9 = 36

You can swap the order too:

*f*(*g*(x)) or (f∘g)(x) means to do *g*(x) first then *f*(x)

*f*(*g*(3)) = *f*(4•3) = *f*(12) = 12^2 = 144

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You can also just combine them into a new singular function:

*g*(*f*(x)) = *g*(x^2 ) = 4•x^2

*f*(*g*(x)) = *f*(4x) = (4x)^2