Calculate functions

Question

Consider the function f(x)=x^2 +4x -1 and g(y)=2y

Calculate the following:

(g/f)(0)

(f/g)(0)

(fοf)(x)

Answer 1

you've ben given: f(x)=x^2 +4x -1 and g(y)=2y

This could have been stated, alternatively, as f(t)=t^2 +4t -1 and g(t)=2t.

 

Can you tell me what f(0) is?  And g(0)?  These calculations are key to beginning the work.

 

(g/f)(0) means g(0) divided by f(0)
(f/g)(0) means f(0) divided by g(0).   Now I am sure that you can answer these two questions successfully.

 

(fοf)(x) is called the composition of f with f and here's how to do it.  It is equivalent to f(f(x)).

Write down f(x) but every time you would have written x, WRITE (x² +4x -1) instead! Then simplify!

Answer 2

First, I will find the composition functions, then plug in.

Then to check I will plug in first and do the operation. 

The results should be the same.

 

(a)  g/f =  (2y)/(x^2 + 4x - 1)

      So (g/f)(0) = (2*0)/(0^2 + 4*0 - 1) = 0

 

(b) f/g = (x^2 + 4x - 1)/(2y(

    So (f/g)(0) = (0^2 + 4(0) - 1)/(2*0) which is undefined because division by zero is not allowed

 

(c) f comp f = (x^2 + 4x - 1)^2 + 4(x^2 + 4x - 1) - 1

 

                  = x^4 +4x^3 - x^2  + 4x^3 + 16x^2 - 4x -x^2 -4x + 1 + 4x^2 + 16x - 4 - 1

 

                 = x^4 + 8x^3 + 18x^2 + 8x - 4

 

And now for the checks:

 

 g(0)/f(0) = 0/-1 = 0 <--- YES

 

f(0)/g(0) = -1/0   <---- UNDEFINED but correct

 

To check the last one let x=0,1

 

f(0) = -1

f(-1) = 1 - 4 - 1 = -4

 

So f(f(0)) = f(-1) = -4

 

Now plugging zero into the function in part c also gives a result of -4, so it checks.

 

To make double sure, f(1) = 4

                              f(4) = 16 + 16 - 1 = 31

 

f(f(1)) = f(4) = 31

Plugging 1 into the function in part c gives a result of 31, so it checks

 

Your answers are shown in parts (a) through (c) above and are double checked for accuracy