How do you find #(f @ g)(x)# and its domain, #(g @ f)(x)# and its domain,# (f @ g)(-2) # and #(g @ f)(-2)# of the following problem #f(x) = x+ 2#, #g(x) = 2x^2#?

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Answer 1 · 2018-01-17T13:33:25

See the explanation below...

By the composition #(f\circ g)(x)#, we mean #f(g(x))# and by #(g\circ f)(x)#, we mean #g(f(x))#.

To find #f(g(x))#, we need to put the value of #g(x)# for every value of #x# in #f(x)#. So, by doing this, we get:

#(f\circ g)(x)=f(g(x))=(2x^2)+2#

#=2x^2+2#

The domain of a function is the set of values for which the function is real and defined.

This above evaluated function has no undefined points. The domain is # -oo < x < oo #

Similarly, we have:

#(g\circ f)(x)=g(f(x))=2(x+2)^2#

Simplify:

#=2x^2+8x+8#

Domain: # -oo < x < oo #

Now, in the same manner, find #(f\circ g)(-2)# as:

#=2(-2)^2+2#

Simplify:

#=10#

And:

#(g\circ f)(-2)=2(-2)^2+8(-2)+8#

Simplify:

#=0#