consider the function f(x)=1-ln(x). Write the expression of (fof)(x) and find the domain of fof.
f(f) = 1 - ln(f) = 1 - ln(1 - ln(x))
the domain of lnx is x > 0
The domain of ln(1-lnx) is 1-lnx > 0, or x < e
so the domain of (fof)(x) is (0,e)
To find the expression of (fof)(x), you need to substitute f(x) into itself. Let's go step by step:
1. Start with the original function: f(x) = 1 - ln(x)
2. Substitute f(x) into f(x): fof(x) = f(f(x))
3. Replace the x in the original function with f(x): f(f(x)) = 1 - ln(f(x))
Now, we need to find the domain of fof(x), which means determining the valid values of x.
1. Start with the domain of f(x), which is the set of values for x where the function is defined. In this case, f(x) = 1 - ln(x) is defined for x > 0 since the natural logarithm function is only valid for positive numbers.
2. Substitute the expression f(x) into fof(x):
fof(x) = 1 - ln(f(x)) = 1 - ln(1 - ln(x))
Now, let's analyze the domain of fof(x):
1. The natural logarithm function (ln) is only defined for positive values, so 1 - ln(x) must be positive. This implies:
1 - ln(x) > 0
ln(x) < 1
2. Taking the exponential of both sides, we get:
e^ln(x) < e^1
x < e
3. Lastly, since both f(x) and ln(x) are defined for x > 0, we know that fof(x) is also defined for x > 0.
Therefore, the domain of (fof)(x) is x < e, where e represents Euler's number, approximately 2.718.