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?

1 Answer
Jan 17, 2018

See the explanation below...

Explanation:

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