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] .