If f(x)=2x+1/x-3 and g(x)=1-x/2
Show that f(g)(x)=g(f)(x)
Before I solve this, clarify if you mean it the way you typed it, or
if you meant:
f(x) = (2x+1)/(x-3) or f(x) = 2x + 1/(x-3) or ...
and
g(x) = (1-x)/2
Hi Reiny,
I did not get any parentheses with the question. I got it the way I typed it.
Thanks.
To show that f(g)(x) is equal to g(f)(x), we need to compute both expressions and compare the results.
First, let's calculate f(g)(x):
Step 1: Substitute g(x) into f(x):
f(g)(x) = f(1-x/2)
Step 2: Replace x in f(x) with 1-x/2:
f(g)(x) = 2(1-x/2) + 1/(1-x/2 - 3)
Step 3: Simplify the expression:
f(g)(x) = 2 - x + 1/(1-x/2 - 3)
Now, let's calculate g(f)(x):
Step 1: Substitute f(x) into g(x):
g(f)(x) = g(2x+1/x-3)
Step 2: Replace x in g(x) with 2x+1/x-3:
g(f)(x) = 1 - (2x + 1/x - 3)/2
Step 3: Simplify the expression:
g(f)(x) = 1 - 2x - 1/x + 3/2
Upon comparing these two expressions, f(g)(x) = g(f)(x) if they are equal.
Simplifying f(g)(x) and g(f)(x) further:
f(g)(x) = 2 - x + 1/(1-x/2 - 3)
g(f)(x) = 1 - 2x - 1/x + 3/2
By examining these expressions, we can see that f(g)(x) is not equal to g(f)(x). Hence, the given statement is false.