(b) Find the exact value of ( )cos p q+ .6. Functions f and g are defined on R by• ( ) 2 5f x x= +• ( ) 2 2g x x x= − .(a) Find an expression for ( )( )f g x .(b) Find an expression for ( )( )g f x .(c) Express ( )( ) ( )( )g f x f g x− in the form ( )2a x b c+ + .123214
Question
(b) Find the exact value of ( )cos p q+ .6. Functions f and g are defined on R by• ( ) 2 5f x x= +• ( ) 2 2g x x x= − .(a) Find an expression for ( )( )f g x .(b) Find an expression for ( )( )g f x .(c) Express ( )( ) ( )( )g f x f g x− in the form ( )2a x b c+ + .123214
Solution
The question seems to be a bit jumbled, but I'll try to answer it as best as I can.
(a) To find an expression for ( )( )f g x , we first need to find the value of g(x) and then substitute it into f(x).
Given that g(x) = 2x^2 - x, we substitute this into f(x) to get:
f(g(x)) = 2(2x^2 - x) + 5 = 4x^2 - 2x + 5
(b) To find an expression for ( )( )g f x , we first need to find the value of f(x) and then substitute it into g(x).
Given that f(x) = 2x + 5, we substitute this into g(x) to get:
g(f(x)) = 2(2x + 5)^2 - (2x + 5) = 8x^2 + 20x + 25 - 2x - 5 = 8x^2 + 18x + 20
(c) To express ( )( ) ( )( )g f x f g x− in the form ( )2a x b c+ + , we first need to find the expressions for g(f(x)) and f(g(x)) and then subtract the latter from the former.
From parts (a) and (b), we have:
g(f(x)) = 8x^2 + 18x + 20 f(g(x)) = 4x^2 - 2x + 5
So, g(f(x)) - f(g(x)) = (8x^2 +
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