Let f (x) = 5x^2 – 9 and g(x) = x – 3.
(a) Find the composite function ( f o g)(x) and simplify. Show work.
(b) Find ( f o g)(2) . Show work.
To find the composite function (f o g)(x), we need to substitute g(x) into f(x) and simplify the expression. The composite function is written as (f o g)(x) or f(g(x)).
(a) Let's find (f o g)(x):
First, we substitute g(x) = x - 3 into f(x):
f(g(x)) = f(x - 3) = 5(x - 3)^2 - 9
To simplify this expression, we need to expand and combine like terms:
f(g(x)) = 5(x^2 - 6x + 9) - 9
= 5x^2 - 30x + 45 - 9
= 5x^2 - 30x + 36
Therefore, the composite function (f o g)(x) is 5x^2 - 30x + 36.
(b) Now, let's find (f o g)(2):
To find (f o g)(2), we substitute x = 2 into the composite function expression we found in part (a):
(f o g)(2) = 5(2)^2 - 30(2) + 36
= 5(4) - 60 + 36
= 20 - 60 + 36
= -4
Therefore, (f o g)(2) = -4.