Given that h(x) = x+4 and g(x) = rootx-1, find each of the following, if it exists.
a. (gh)(10)
b. (h/g)(1)
c. (hg)(3)
To find the value of each expression, we need to substitute the corresponding values of x into the given functions and perform the required operations. Let's start by evaluating each expression step by step:
a. (gh)(10):
To find (gh)(10), we first need to evaluate g(10), then h(g(10)), and finally substitute x = 10 into the h(x) function.
1. g(10):
Using the given function g(x) = √(x - 1), we substitute x = 10:
g(10) = √(10 - 1) = √9 = 3.
2. h(g(10)):
Now, we substitute g(10) = 3 into the h(x) function:
h(g(10)) = h(3) = 3 + 4 = 7.
Therefore, (gh)(10) = 7.
b. (h/g)(1):
To find (h/g)(1), we first evaluate h(1), then g(1), and finally divide h(1) by g(1).
1. h(1):
Using the given function h(x) = x + 4, we substitute x = 1:
h(1) = 1 + 4 = 5.
2. g(1):
Using the given function g(x) = √(x - 1), we substitute x = 1:
g(1) = √(1 - 1) = 0.
3. (h/g)(1):
Finally, we divide h(1) by g(1):
(h/g)(1) = 5 / 0.
However, division by zero is undefined. Therefore, (h/g)(1) does not exist.
c. (hg)(3):
To find (hg)(3), we first evaluate h(3), then g(h(3)), and finally substitute x = 3 into g(x).
1. h(3):
Using the given function h(x) = x + 4, we substitute x = 3:
h(3) = 3 + 4 = 7.
2. g(h(3)):
Now, substitute h(3) = 7 into the g(x) function:
g(h(3)) = g(7) = √(7 - 1) = √6.
Therefore, (hg)(3) = √6.