Verify the function f and g, are inverses of each other by showing that f(g(x))=x and g(f(x))=x Graph the functions.
f(x)=6/(1-x)
g(x)=(1-6/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 whether the functions f and g are inverses of each other, we need to show that when we apply one function after the other, it results in the original input value.
1. To check if f(g(x)) = x:
First, substitute g(x) into f(x):
f(g(x)) = f(1-6/x)
Now, replace x in f(x) with (1-6/x):
f(1-6/x) = 6 / (1 - (1-6/x))
Simplify further:
f(1-6/x) = 6 / (1 - 1 + 6/x)
f(1-6/x) = 6 / (6/x)
f(1-6/x) = 6 * (x/6)
f(1-6/x) = x
Since f(g(x)) = x, the first condition is satisfied.
2. To check if g(f(x)) = x:
Now, we substitute f(x) into g(x):
g(f(x)) = g(6/(1-x))
Replace x in g(x) with (6/(1-x)):
g(6/(1-x)) = 1 - 6 / (6/(1-x))
Simplify further:
g(6/(1-x)) = 1 - 6 * (1-x) / 6
g(6/(1-x)) = 1 - (1 - x)
g(6/(1-x)) = 1 - 1 + x
g(6/(1-x)) = x
Since g(f(x)) = x, the second condition is also satisfied.
By verifying that f(g(x)) = x and g(f(x)) = x, we can confirm that the functions f and g are inverses of each other.
To graph the functions, we can plot their respective equations on a coordinate plane. However, due to the complexity of the equations, it may be challenging to do so using a simple line graph.