verify that the functions of f and g are inverse of each other by showing that f(g(x))=x and g(f(x))=x; f(x)=In(x-1),g(x)=1+e^x
what's the problem? Just plug and chug:
f(g) = ln(g-1) = ln(1+e^x-1) = ln(e^x) = x
g(f) = 1+e^f = 1 + e^(ln (x-1)) = 1 + x-1 = x
Just keep in mind the definition of a logarithm:
e^lnx = x
ln e^x = x
To verify that two functions, f and g, are inverses of each other, we need to show that f(g(x)) = x and g(f(x)) = x.
Let's start by finding f(g(x)).
Given f(x) = ln(x - 1) and g(x) = 1 + e^x, we replace x in f(x) with g(x):
f(g(x)) = ln(g(x) - 1)
Substituting g(x) = 1 + e^x into the above equation gives:
f(g(x)) = ln(1 + e^x - 1)
Simplifying further:
f(g(x)) = ln(e^x)
Using the property that ln(e^x) = x, we have:
f(g(x)) = x
Thus, we have shown that f(g(x)) = x.
Now, let's find g(f(x)).
Using g(x) = 1 + e^x, we replace x in g(x) with f(x):
g(f(x)) = 1 + e^(f(x))
Substituting f(x) = ln(x - 1) into the above equation gives:
g(f(x)) = 1 + e^(ln(x - 1))
Simplifying further:
g(f(x)) = 1 + (x - 1)
g(f(x)) = x
Therefore, we have also shown that g(f(x)) = x.
Since both f(g(x)) = x and g(f(x)) = x, we can conclude that f and g are inverse functions.