Let f(x)=x^2+x and g(x)=2x+1. Find and simplify the following:

Question

a) (g o f)(x)
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b) (f o g)(x) -- the o is colored in.
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c) (f+g)(2)
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d) (f-g)(3)

Answer 1

f (x ) = x ^2 + x and g (x ) = 2x +1

a) ( g o f  )(x ) = g ( f (x ))

Substitute the expression for functioning f (in this case x ^2 +x ) for f (x ) in the composition.

= g (x ^2 + x )

Now substitute this expression (x ^2 + x) in to function g in place of the x  value.

= 2(x ^2 + x ) + 1

( g o f )(x ) = 2x ^2 + 2x +1.

 

b ) ( f o g )(x ) = f ( g (x ))

Substitute the expression for functioning g (in this case 2x +1 ) for g (x ) in the composition.

= f (2x + 1)

Now substitute this expression (2x + 1) in to function f  in place of the x  value.

= (2x + 1)^2 + (2x + 1)

= 4x ^2 + 1 + 4x + 2x + 1

( f o g )(x )  = 4x ^2 + 6x + 2.

 

c ) ( f + g )(2) = [f (x ) + g (x )](2)

f (x ) + g (x) =  (x ^2 + x ) + (2x + 1)

= x ^2 + x + 2x + 1

= x ^2 + 3x + 1

( f + g ) (2) = (2)^2 + 3(2) + 1

= 4 + 6 + 1

(f + g )(2) = 11.

 

d ) (f - g )(3) = [f (x ) - g (x)](3)

f (x ) - g (x) = (x ^2 + x ) - (2x + 1)

= x ^2 + x - 2x -1

= x ^2 - x -1

( f - g )(3) = (3)^2 - 3 - 1

 = 9 - 4

( f - g )(3) = 5.