Let f(x)=8x and g(x)=x/8, how do you find each of the compositions and domain and range?

1 Answer
Nov 24, 2015

f(g(x))=xcolor(white)("XXX")andcolor(white)("XXX")g(f(x))=x
Both functions and their compositions have Domains and Ranges of (-oo,+oo)

Explanation:

Sometimes the use of x in multiple definitions can cause confusion, so let's re-write the base equations as:
color(white)("XXX")f(a)=8a
and
color(white)("XXX")g(b)=b/8

So, replacing a with g(b) we have
color(white)("XXX")f(g(b)) = 8*g(b) = 8*b/8 = b
(or, using the original x as the variable: f(g(x))=x)

Similarly g(f(x)) = x

{ (f(x) = 8x), (g(x) = (x)/(8)), (f(g(x))=x), (g(f(x)))=x :}
color(white)("XXX")are all defined for all values of x
and
therefore have Domains of all Real values, (-oo,+oo)

Replacing, for example f(x) with y
we can see that f(x)=8x <=>y/8=x
which is defined for all Real values of y.
(this process can be done for all of the functions)

So the Ranges of these functions is also all Real values (-oo,+oo)