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# Composite function question

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A)

B)Help would be very much appreciated!

Dec 30, 2017

#1
+8394
+1

Question 1

a)     f( x )  =  3x2 - 5x + 6    and    g(x)  =  x2 + 3x

f o g   =   f( g(x) )

f o g   =   f( x2 + 3x )

f o g   =    3( x2 + 3x )2 - 5( x2 + 3x ) + 6    I'll let you simplify these if necessary...

The domain is (-∞, ∞), and from this graph, we can see that the range is  [47/12, ∞) .

g o f   =   g( f(x) )

g o f   =  g( 3x2 - 5x + 6 )

g o f   =   ( 3x2 - 5x + 6 )2 + 3( 3x2 - 5x + 6 )

The domain is  (-∞, ∞), and from this graph, we can see that the range is  [3901/144, ∞) .

b)     f(x)  =  2x    and    g(x)  =  3 - x

f o g   =   f( g(x) )

f o g   =   f( 3 - x )

f o g   =   23 - x

The domain is  (-∞, ∞), and the range is  (0, ∞) .

g o f   =   g( f(x) )

g o f   =   g( 2x )

g o f   =   3 - 2x

The domain is  (-∞, ∞), and the range is  (-∞, 3) .

c)     f(x)  =  sin x    and    g(x)  =  x

f o g   =   f( g(x) )

f o g   =   f(  x  )             We already know that  f(x)  =  sin x , so....

f o g   =   sin x

The domain of this is  (-∞, ∞), and the range is  [-1, 1] .

g o f   =   g( f(x) )

g o f   =   g( sin x )     Plug in  " sin x "  for  " x "  into the function  g(x) .

g o f   =   sin x

The domain is  (-∞, ∞),  and the range is  [-1, 1] .

Dec 30, 2017
edited by hectictar  Dec 30, 2017
#2
+8394
+1

Question 2

a)    3 - 3x   =   3 - f(x)   =   g( f(x) )   =  g o f

b)    x2 - 6x + 6   =   x2 - 6x + 9 - 3   =   (3 - x)2 - 3   =   ( g(x) )2 - 3   =   h( g(x) )   =   h o g

c)    6 - x2   =   3 + 3 - x2   =   3 - x2 + 3   =   3 - (x2 - 3)   =   3 - ( h(x) )   =   g( h(x) )   =   g o h

Dec 30, 2017