SOLUTION: For each pair of functions f and g below, find f (g (x)) and g (f(x)) Then, determine whether f and g are inverses of each other. Simplify your answers as much as possible. (Ass

Question 1189760: For each pair of functions f and g below, find f (g (x)) and g (f(x))
Then, determine whether f and g are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all x in the domain of the composition.
You do not have to indicate the domain.)
(a) f(x) =1/4x , x ≠ 0
g(x) = 1/4x , x ≠ 0
f(g(x))=
g(f(x))=
are f and g inverses of each other? yes or no?

(b) f(x)=x+6
g(x)=x+6
f(g(x))=
g(f(x))=
are f and g inverses of each other? yes or no?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if f(x) is the inverse of g(x), then f(g(x)) would be equal to x.
if g(x) is the inverse of f(x), then g(f(x)) would be equal to x.

(a) f(x) = 1/4 * x , x ≠ 0
g(x) = 1/4 * x , x ≠ 0
f(g(x))= 1/4 * (1/4 * x) = 1/16 * x which is not equal to x.
g(f(x))= 1/4 * (1/4 * x) = 1/16 * x which is not equal to x.
are f and g inverses of each other? yes or no?
answer is no.
in fact, these equations are identical to each other.

(b) f(x)=x+6
g(x)=x+6
f(g(x))= (x + 6) + 6 = x + 12 which is not equal to x.
g(f(x))= (X + 6) + 6 = X + 12 which is not equal to x.
answer is no.
in fact, these equations are identical to each other.

you will notice that f(x) and g(x) are the same equation, i.e. they are identical.
as such, they can't possibly be inverses of each other.

here's a good reference on inverse functions.

https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut32b_inverfun.htm

note that, if your equation doesn't pass the horizontal line test, then it is not a function.
in that case, however, you can turn it into a function by restricting the domain.
for example, in example 4 of the tutorial, if you restricted the domain to all values of x >= 0, then the equation would be a function because it would then pass the horizontal line test .

to find the inverse function of f(x0 = x + 6, you would do the following:
let y = f(x) to get y = x + 6.
replace y with x and x with y to get x = y + 6.
solve for y to get y = x - 6.
f(x) = x + 6 is the original functiojn.
g(x) = x - 6 is the inverse fucntion.
f(g(x)) = (x - 6) + 6 which is equal to x.
this passes the test.

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