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) i(x)=(x-2) • g(x)

(h° 3f ° 2i ÷ g)(x) for x= -4, -3, 2

Answer 1

The key to this problem is to identify the functions; isolate the functions dependent on only x; and define each function in terms of x. Once every function is in terms of x then we can apply them to the equation that needs to be solved.

1. Identify the functions: Here we are given four functions f(x); g(x); h(x); and i(x) so anytime we see these letters appear in an equation is where we would substitute the function.
2. Isolate the functions dependent on only x: At a glance we can tell that only g(x) is purely in terms of x, every other function contains another function within their equations.
3. Define each function in terms of x:
4. Since g(x) is in terms of x and i(x) contains g(x) then we can substitute 3x+4 for g(x) making i(x)=(x-2)*(3x+4) which then simplifies to i(x)=3x²-2x-8
5. Since h(x) contains i(x) then we can substitute the i(x) we made in 3a to get h(x)=3x²-11-(3x²-2x-8) which then simplifies to h(x)=2x-3
6. Since f(x) contains g(x) and h(x) then we can substitute those equations to get f(x)=2(3x+4)-(2x-3) which then simplifies to f(x)=4x+11
7. Now we apply our defined equations to the equation in question. There are two ways to do this, we can either apply our x values to each individual equation first or make the equation in question in terms of x. In this case I will make the equation in terms of x first. The first part of this process is understanding that the ° symbol does not represent degrees in this case, it represents the substitution of a function into another function. For example, if you saw (f°g)(x) then you would take the equation for g(x) and substitute it for every x value in the f(x) equation.
8. Based on the process around ° it would be best to start from i(x) and work backwards to f(x) then h(x). So 2i=2(i(x))=2(3x²-2x-8)=6x²-4x-16
9. Ignoring the 3 we then evaluate f°2i=f(2(i(x)))=4(6x²-4x-16)+11=24x²-16x-64. Then we multiply the 3 to get 3f°2i=3(24x²-16x-64)=72x²-48x-192
10. Then we get to h°3f°2i=2(72x²-48x-192)-3=144x²-96x-384 which can simplify to 48(3x²-2x-8) or 48(3x+4)(x-2).
11. Finally we can divide g(x) making (h°3f°2i÷g)(x)=(48(3x+4)(x-2))/(3x+4)=48(x-2)
12. With the final equation in terms of only x we can finally solve by substituting each x value, giving us -288 when x=-4; -240 when x=-3; and 0 when x=2.