can somone help me with this problem.

Question

 

Let f(x) = 6x²+7 and g(x) = x-3

 

a) Find the composite function (f°g)(x) and simplify.

 

b) Find (f°g)(-1)

 

Okay so if I am understanding this correctly the formula would change to f(g(x)) and part b would be f(g(-1))

 

But I dont really know how i would input that. Would it be 6x²+7(x-3(x))? and the second part be the same except x would be -1? I am not sure how I would solve this or if i am even on the right track.

Answer 1

For Part A

So a composite function is where you essentially combine two different equations. (f^og)(x) can also be written as:

 

f[g(x)]

 

Now, we know that g(x) = x - 3, so we can also write our composite function like this:

 

f[x - 3]

 

This means that wherever there is an "x" in the function f, we should substitute "x - 3." So the function becomes:

 

f(x - 3) = 6(x - 3)² + 7

 

Now we just need to simplify by using FOIL and adding like terms:

 

f(x - 3) = 6(x² - 6x + 9) + 7

f(x - 3) = 6x² - 36x + 54 + 7

f(x - 3) = 6x² - 36x + 61

OR

(f^og)(x) = 6x² - 36x + 61

 

For Part B

You have the right idea for this part, x would be equal to -1, but only for the g function. Since the composite function can also be written as f[g(x)], we should first input -1 for x in function g:

 

g(x) = x - 3

g(-1) = -1 - 3

g(-1) = -4

 

Now, we substitute -4 for x in function f:

 

f(x) = 6x² + 7

f(-4) = 6(-4)² + 7

f(-4) = 6(16) + 7

f(-4) = 96 + 7

f(-4) = 103

OR

(f^og)(-1) = 103

 

Another way to find this same answer, is simply to plug in -1 to the composite function we found in Part A:

 

(f^og)(-1) = 6x² - 36x + 61

(f^og)(-1) = 6(-1)² - 36(-1) + 61

(f^og)(-1) = 6 + 36 + 61

(f^og)(-1) = 103

 

 

If you would like to look at more examples, here is an excellent link:

http://tutorial.math.lamar.edu/Classes/Alg/CombineFunctions.aspx#Fcns_Comb_Ex3_e