Given f(x) = 1/x and g(x) = 2x + 6, now calculate the following:

Question

f(x) = 1/x and g(x) = 2x + 6, calculate the following:

a) f ◦ g (x) = ? , its domain is all real numbers except: ?

b) g ◦ f (x) = ? , its domain is all real numbers except: ?

c) f ◦ f (x) = ? , its domain is all real numbers except: ?

d) g ◦ g (x) = ? , its domain is ( ? , ? ).

Answer 1

When faced with composition of functions, it is usually less confusing

when the two functions have different variable names.

So let's define

F(u) = 1/u

G(u) = 2u + 6

a)

To calculate f◦g(x), we can just as well calculate F◦g(x).

And that is obtained by substituting 2x + 6 for any occurrence of u in the definition of F(u):

f◦g(x) = f(g(x)) = F(g(x)) = 1/(2x+6)

As you said, the domain is all real numbers, except -3.

b)

g◦f(x) = G(f(x)) = 2/x + 6

As you said, the domain is all real numbers, except 0.

c)

f◦f(x) = F(f(x))= 1/(1/x)) = x

As you said, the domain is all real numbers, except 0.

The reason is a little different than in (b):

f(x) being undefined at 0, prevents us from evaluating h(f(x)) for any function h,

regardless of any simplification.

d)

g◦g(x) = G(g(x)) = 2(2x + 6) + 6 = 4x + 18

The domain is all real numbers, i.e. (-∞, +∞).