Show that f and g are inverse functions...(Please show and list all steps)?

Question

f(x) =(x-1/x-5)

g(x)= -(5x+1\x-1)

Answer 1

Sam, you got to be more careful with your notation. Given f(x) = (x - 1)/(x - 5), the inverse will be a function g(x) such that f(g(x)) = x. Now, f(g(x)) = [g(x) - 1]/[g(x) - 5] = x. Solving for g(x), we have g(x) - 1 = x*g(x) - 5x, so g(x)*(1 - x) = 1 - 5x, so g(x) = (1 - 5x)/(1 - x). (See the difference?)

As a check, we show that g(f(x)) = x. We have g(f(x)) = [1 - 5*f(x)]/[1 - f(x)] = [1 - 5*{(x - 1)/(x - 5)}]/[1 - (x - 1)/(x - 5)] = [x - 5 - 5*(x - 1)]/[x - 5 - x + 1] = [-4x/(-4)] = x.

In general, here is what is involved in the inverse-function relation: the inverse of a one-to-one function f(x) is a function g(x) such that f(g(x)) = x and g(f(x)) =x. There are also some requirements that the domain of f = the range of g and the domain of g = the range of f. (If "one-to-one," "range," and "domain" are strange terms, fuggedaboutit. Just concentrate on f(g(x)) = x = g(f(x)).)

Answer 2

maybe I'm missing something, but I don't find them to be inverses.
starting with the first function:
f(x) =(x-1/x-5)
Write the function as an equation:
y = (x-1/x-5)
solve the equation for x:
y(x-5) = x-1)

xy-5y = x-1

xy-x = 5y-1

x(y-1) = 5y-1

x=5y-1/y-1

x = (5y-1)/(y-1)

x = -(-5y+1)/(y-1)

or written as a function:

g(x) =-(-5x+1)/(x-1)

which is not the function that we wanted!!

Answer 3

f(x) = (x-1)/(x-5)
y = (x-1)/(x-5)
x = (y-1)/(y-5)
xy - 5x = y-1
xy - y = 5x -1
y(x-1) = 5x -1
y = (5x - 1) / (x - 1) = g(x)
there f(x) is the inverse function of g(x)

Answer 4

I can't because they are not inverse functions.

Just take x = 3
f(3) = -1
g(-1) = -2
It should have been 3 if g were the inverse of f.

Answer 5

f(x) = y = (x-1)/(x-5)

y(x-5) = x-1

xy - 5y = x - 1

xy - x = 5y - 1

x(y - 1) = 5y-1

x = 5y-1/y-1

= -(5y+1)/y-1

replacing y with x

f^-1(x) = -(5x+1)/(x-1) = g(x)

so f and g are inverse functions