Question 1170515: f and g are one to one functions and satisfy
f(3)=7,f(11)=16,f(7)=11,f(16)=3,g(7)=16,g(11)=7,g(3)=11,g(16)=3
Find
((f∘g)(16))−1−((g∘f)(16))−1
Answer:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step:
**1. Understand Composition of Functions**
* (f∘g)(x) = f(g(x))
* (g∘f)(x) = g(f(x))
**2. Calculate (f∘g)(16)**
* (f∘g)(16) = f(g(16))
* We're given g(16) = 3
* So (f∘g)(16) = f(3)
* We're given f(3) = 7
* Therefore, (f∘g)(16) = 7
**3. Calculate (g∘f)(16)**
* (g∘f)(16) = g(f(16))
* We're given f(16) = 3
* So (g∘f)(16) = g(3)
* We're given g(3) = 11
* Therefore, (g∘f)(16) = 11
**4. Find ((f∘g)(16))−1**
* ((f∘g)(16))−1 = 7−1
* We want to find the inverse of 7.
* To find f^-1(7), we look for a value x such that f(x)=7. f(3)=7. therefore f^-1(7) = 3.
* To find g^-1(11), we look for a value x such that g(x)=11. g(3)=11. therefore g^-1(11)=3.
* ((f∘g)(16))−1 = f^-1(7) = 3
**5. Find ((g∘f)(16))−1**
* ((g∘f)(16))−1 = 11−1
* To find f^-1(11), we look for a value x such that f(x)=11. f(7)=11. therefore f^-1(11) = 7.
* To find g^-1(11), we look for a value x such that g(x)=11. g(3)=11. therefore g^-1(11)=3.
* ((g∘f)(16))−1 = g^-1(11) = 3.
**6. Calculate the Final Result**
* ((f∘g)(16))−1 − ((g∘f)(16))−1 = 3 - 3 = 0
**Final Answer:** The final answer is 0.
|
|
|
