evaluate the composite function for the indicated values of x:

Question

f(x)= (2g-h)(x) g(x)= 3x+4 h(x)= 3x²-11-i(x)

(f º g º h)(x) for x= -2, 1, 4

Answer 1

Mona, my heart is with you.

Assuming i(x) means x*Sqrt(-1)

This is certainly complicated. Let’s start by expressing f(x) in terms of x only.

Plug (3x+4) into the formula for f in place of g.

Also plug 3x2-11-i(x) in place of h.

f(x) = 2*(3x+4) - (3x2-11-i(x))=

6x+4 - 3x² +11 +i(x)=

-3x² +(6+i)x+ 15

Now we are ready to perform the composition of functions

Work from right to left: what is g º h?

g(h) = 3*h(x)+4=

3*(3x²-11-i(x)) + 4=

9x² -33 -3i*x + 4 =

[call this A] g(h(x))=9x² -29 -3i*x

Now, what is 

(f º g º h)(x)?

 f(A) = -3A²+(6+i)A + 15

so substitute the expression for g in for “A”

[call this B]= 

-3(9x² -29 -3i*x)²+(6+i)*(9x²-29-3i*x) +15

Finally substitute the different values into [B] for x: -2, 1, 4

I’ll do 1 for you

-3(9 -29-3i)2 + (6+i)*(9-29-3i)+15

= -3(-20-3i)2+ (6+i)*(-20-3i) +15

= -3(400 +120i +9i2) + (-120-38i-3i2) + 15

= -1200 -360i +27  -120 -38i +3 +15

= (-1200+27-120+18) -398i

= (-1320+45) -398i

(Second answer) = -1275 - 398i

Now plug -2 into B for the first answer.

Then plug 4 into B for the third answer

I hope this formula cures the common cold.

If you have any questions, please ask.