Section 1.4- Combinding Of Functions

We all have learned the basic of functions at this point so know we're going to take it to a new level. next we're going to try combining them. in this blog I'll show you 4 of the five basic ways to do this. starting with the basic function f(x)=2x-3 and g(x)=x^2 -1. now remember these functions because we're going to come back to them quite a lot.


1.)We'll start with the sum of two functions. taking our to functions f(x) and g(x) we are going to create (f+g)(x) this is like saying f(x) + g(x) but you must write it as the first example as to avoid confusion.
lets start.
You've got your function? f(x)=2x-3 and g(x)=x^2 -1? now add them like this...
(f+g)(x)=(2x-3) + (x^2 -1) simply
=x^2 + 2x - 4

here a a few more example just in case your still shaky...
starting with f(x)= 2x-1 and g(x)= x^2 + 2x -1 add...
(f+g)(x)= f(x)+ g(x)
=(2x-1) + (x^2 + 2x -1) simply
=x^2 + 4x
Starting with f(x)= x^3+8 and g(x)= x+1 add...
(f+g)(x)= f(x) + g(x)
=(x^3+8) + (x+1)
=x^4 + x^3 + 8x + 9


2.)Now for finding the difference (aka. subtracting) going back to our two orignal fuctions f(x) and g(x) we are going to create (f-g)(x) this is like saying f(x) - g(x).
OK, You've got your function? f(x)=2x-3 and g(x)=x^2 -1? now subtract them like this...
(f-g)(x)=(2x-3) - (x^2 -1) simply
=-x^2 + 2x - 2

And now for so more examples....
starting with h(x)= x^2-4 and j(x)= x^2 + 3x
(h-j)(x)=(x^2-4)-(x^2 +3x)
=x^2-4-x^2-3x
=3x-4
Starting with k(x)=3x+7 and d(x)= x^2 -3
(k-d)(x)=(3x+7)-(x^2-3)
=3x+7-x^2+3
=-x^2+3x+10


3.) Now we'll find the Product of two functions or in other words well multiply them.
back to our functions f(x) and g(x) we're going to find (fg)(x) or f(x) * g(x)...
take (fg)(x)=(2x-3)(x^2-1)
=2x^2 - 3x^2 - 2x + 3

More examples...
Starting with t(x)=x^2 and u(x)= x -3
(tu)(x)=(x^2)(x-3)
=x^3 - 3x^2

4.) and finally finding the quotient or dividing the functions. we're going to take f(x) and g(x) and make (f/g)(x) or f(x)/g(x).
Take (f/g)(x)=(2x-3)/(x^2-1)
for these that's all you need to do however typically because of how simple that is people might ask you to find the domain...
on this example it would be x =/= ±1

more examples...
starting with w(x)=√x and y(x)=√4-x^2
(w/y)(x)= (√x)/(√4-x^2)
the domain would be (w/y)(x) would be [0,2)