Need help with two function questions. Let f(x)=9+x+x^2. 1. Find f(x + h). 2. Find f(x+h)-f(x)/h and simplify

Question

Please help me solve these two equations showing step-by-step instructions. I have attempted these several times, but have not been successful. 

 

Let f(x)=9+x+x^2.

 

1. Find f(x + h) and simplify.
2. Find f(x+h)-f(x)/h and simplify.

 

Thank you.

Answer 1

Start by substituting (x+h) into the equation for (x):

 

f(x+h)=9+(x+h)+(x+h)^2

 

Then simplify the equation using the FOIL method on (x+h)^2:

 

f(x+h)=9+x+h+x^2+2hx+h^2

 

This answers part one of the question because you cannot simply it any further.

 

 

 

Then using the answer from above, write out the equation for part two:

 

 

                          this is the answer from part 1      this is the original problem

f(x+h)-f(x)/h=      [9+x+h+x^2+2hx++h^2]          -         [9+x+x^2]

                           ______________________________________________

 

                                                               h

 

So you have the answer from part one f(x+h) minus the original equation, f(x), all divided by h.

 

Now simplify the equation by first distributing the minus sign through the equation:

 

f(x+h)-f(x)/h=    [9+x+h+x^2+2hx+h^2]  + [-9-x-x^2]

                         __________________________________

                                                   h

 

Now you can see that the 9 and -9 cancel, the x and -x cancel and the x^2 and -x^2 cancel leaving:

 

f(x+h)-f(x)/h=   h+2hx+h^2

                       ___________

                              h

 

Now divide by h and you have left:

 

f(x+h)-f(x)/h=  1+2x+h

 

 

 

 

 

 

Answer 2

f(X) = X² + X +9

f(X+h) = ( x+h)² + (X + h) + 9

            =  X² + 2hx + h² + X +h +9

             = X² +(2h+1)X +h +9

[ f (X + h)- f (X)] /h = ( X² +2hx+X+h+9- X² -X - 9)/h =

                                         2hx+h/ h = 2x +1 

Answer 3

1. For this one, instead of using "x" in our expression for f(x), we're using "x+h".  So to find f(x+h), plug in (x+h) instead of x in your expression.  The unsimplified version should look something like

      f(x+h) = 9 + (x+h) + (x+h)².

From here, I'll leave the simplification up to you: find like terms (x²'s, x's, etc) and group them together to get a "neat" looking equation.  You'll have to FOIL that squared term.

 

2. This should be straightforward once you concretely know what f(x+h) is.  However as a warning, to fully simplify, you'll have to use factoring and FOILing (expanding things like (x+h)² ).  Hope this helps!