Answer:
To verify that functions f and g are inverse functions, we need to show that applying one function followed by the other will result in the original input. In other words, we need to show that f(g(x)) = x and g(f(x)) = x.
(a) Algebraically:
Let's first find the composition of f(g(x)):
f(g(x)) = f(1 - x/x)
Substituting g(x) into f(x):
f(g(x)) = f(1 - x/x) = 1/(1 + (1 - x/x))
Simplifying the expression inside the parentheses:
f(g(x)) = 1/(1 + (1 - 1)) = 1/(1 + 0) = 1/1 = 1
Now let's find the composition of g(f(x)):
g(f(x)) = g(1/(1 + x))
Substituting f(x) into g(x):
g(f(x)) = g(1/(1 + x)) = 1 - (1/(1 + x))/(1/(1 + x))
Simplifying the expression inside the parentheses:
g(f(x)) = 1 - (1/(1 + x))/(1/(1 + x)) = 1 - 1 = 0
Since f(g(x)) = 1 and g(f(x)) = 0, we can see that f and g are not inverse functions algebraically.
(b) Graphically:
To determine if f and g are inverses graphically, we can plot the graphs of both functions and see if they are reflections of each other over the line y = x.
Graph of f(x) = 1/(1 + x):
```
|
1| ------
| /
| /
| /
| /
| /
| /
| /
| /
0|_/_____________
|
0
```
Graph of g(x) = 1 - x/x:
```
|
1| ----
| \
| \
| \
| \
| \
| \
| \
| ----
0|______________/
|
0
```
Looking at the graphs, we can see that they are not reflections of each other over the line y = x. Therefore, graphically, f and g are not inverse functions.
In conclusion, both algebraically and graphically, f and g are not inverse functions.