For the given functions
f(x) = ln(x) ,
g(x) = |x| - 7,
h(x)=5x+6
find f o g o h and state the exact domain of f o g o h. P
lease show all of your work.
To find f o g o h (read as "f composed with g composed with h"), we need to first evaluate h(x), then substitute the result into g(x), and finally substitute the result into f(x).
1. Evaluating h(x):
h(x) = 5x + 6
2. Substituting h(x) into g(x):
g(h(x)) = |h(x)| - 7
g(h(x)) = |5x + 6| - 7
3. Substituting g(h(x)) into f(x):
f(g(h(x))) = f(|h(x)| - 7)
f(g(h(x))) = f(|5x + 6| - 7)
f(g(h(x))) = ln(|5x + 6| - 7)
Now, let's determine the exact domain of f o g o h.
The domain of f(x) = ln(x) is x > 0, meaning x should be greater than zero.
For g(x) = |x| - 7, there are no restrictions on the domain since any value of x can be plugged into the absolute value function.
Lastly, for h(x) = 5x + 6, there are no restrictions on the domain either, as it is a linear function.
Therefore, the domain of f o g o h is the intersection of the three domains: x > 0.