Composition of Functions

Question

Find f(x+h)-f(x)/h for f(x)=2x^2 + 3x

 

The answer is 4x + 3 + 2h, but I don't know how to arrive at it. I've been always baffled by this type of problem. 

Answer 1

f(x) = 2x^2 + 3x

 

I am assuming you are able to find , say f(3) which is value of f(x) when x = 3.

 

The way you do it is you replace every occurrence of x with 3.So f(3) = 2(3)^2 + 3(2) 

 

So if you are asked about f(x+h), you need to replace every occurrence of x with x+h.

So f(x+h) = 2(x+h)^2 + 3(x+h)

               = 2( x^2 + 2xh + h^2 ) + 3x +3h --I expanded (x+h)^2 using the formula expansion for (a+b)^2

                = 2 x^2 + 4xh +2h^2 + 3x + 3h

 

     f(x)      = 2x^2                       + 3x   -- rewriting f(x)

 

So f(x+h) -f(x)  =     4xh + 2h^2      + 3h     -- I substracted so 2x^2 and 3x got cancelled

 

                      = h (4x + 2h + 3)  --- See all the terms have h as a factor

 

 So[ f(x+h) - f(x)] / h = h (4x + 2h + 3)/ h = 4x +3 +2h  ... h gets cancelled from both numerator and denominator

 

           

Answer 2

So to start, it helps to have the parenthesis correct. Specifically this is [f(x+h)-f(x)]/h.

 

If f(x) = 2x^2+3x, then f(x+h) = 2(x+h)^2+3(x+h) = 2x^2 + 2h^2 + 2*2xh +3x + 3h

 

Thus f(x+h) - f(x) = 2x^2 + 2h^2 + 2*2xh +3x + 3h - 2x^2 - 3x

=2h^2+4xh+2h.

 

When divided through by h as in the original line, it is easy to see where you get 4x + 3 + 2h.