For the given functions f, g, and h, find fogoh and state the exact domain of fogoh. Please show all of your work.
f(x)=lnx
g(x)=|x|-2
h(x)=4x-7
domains:
f: x>0
g: all reals
h: all reals
f(x) = lnx
f(g(h)) = ln(g(h))
= ln(|h|-2)
= ln(|4x-7|-2)
ln(u) is defined only for u>0
so, we need |4x-7|-2 > 0
|4x-7| > 2
so, if 4x-7>=0 (or x>=7/4), 4x-7 > 2 ==> x > 9/4
or, if 4x-7 < 0 (or x<7/4), -(4x-7) > 2 ==> x < 5/4
so, domain is (-oo,5/4) U [9/4,oo)
To find fogoh, we need to follow the order of operations and substitute each function into the previous one until we have the final composition.
Step 1: Find fog
Start with the function f(x) = lnx and substitute g(x) = |x|-2 into f(x).
fog(x) = f(g(x)) = f(|x|-2) = ln(|x|-2)
Step 2: Find fogoh
Now, substitute h(x) = 4x-7 into fog(x).
fogoh(x) = fog(h(x)) = fog(4x-7) = ln(|4x-7|-2)
To determine the exact domain of fogoh, we need to be mindful of any restrictions imposed by the natural logarithm (ln) function and the absolute value function.
For the natural logarithm, the argument (the quantity inside the ln function) must be positive. Therefore, |4x-7|-2 > 0.
Next, consider the absolute value. The expression |4x-7| is always positive or zero.
Combining these two conditions, we have |4x-7| > 2.
To solve this inequality, we can write it as two separate inequalities:
1) 4x-7 > 2
2) 4x-7 < -2
Now, solve for x in each inequality:
1) 4x > 9
x > 9/4
2) 4x < 5
x < 5/4
Since the absolute value function results in a positive value, the domain of fogoh is the intersection of the domains of the individual functions. Therefore, the exact domain of fogoh is (5/4, 9/4).