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Verify that f and g are inverses function (a) algebraically (b) graphically

f(x)= 1/1+x , x ≥ 0
g(x)= 1-x/x, 0 < x≤ 1

User Huso
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1 Answer

3 votes

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.

User Kbjr
by
8.0k points

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