If `f : R to R`, `f(x)=x^(2)` and `g(x)=2x+1`, then

Options

A. `fog=gof`
B. `fog ne gof`
C. `fof=gog`
D. none of these

Answer

To solve the problem, we need to find the compositions of the functions \( f \) and \( g \) defined as follows:

- \( f(x) = x^2 \)
- \( g(x) = 2x + 1 \)

We will compute \( f(g(x)) \), \( g(f(x)) \), \( f(f(x)) \), and \( g(g(x)) \).

### Step 1: Calculate \( f(g(x)) \)

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):

\[ f(g(x)) = f(2x + 1) = (2x + 1)^2 \]

Now, we expand \( (2x + 1)^2 \):

\[ (2x + 1)^2 = 4x^2 + 4x + 1 \]

So,

\[ f(g(x)) = 4x^2 + 4x + 1 \]

### Step 2: Calculate \( g(f(x)) \)

Next, we compute \( g(f(x)) \):

\[ g(f(x)) = g(x^2) = 2(x^2) + 1 = 2x^2 + 1 \]

### Step 3: Calculate \( f(f(x)) \)

Now, we find \( f(f(x)) \):

\[ f(f(x)) = f(x^2) = (x^2)^2 = x^4 \]

### Step 4: Calculate \( g(g(x)) \)

Finally, we calculate \( g(g(x)) \):

\[ g(g(x)) = g(2x + 1) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3 \]

### Summary of Results

1. \( f(g(x)) = 4x^2 + 4x + 1 \)
2. \( g(f(x)) = 2x^2 + 1 \)
3. \( f(f(x)) = x^4 \)
4. \( g(g(x)) = 4x + 3 \)

### Conclusion

Now, we can compare the results:

- \( f(g(x)) \) and \( g(f(x)) \) are not equal, as \( 4x^2 + 4x + 1 \neq 2x^2 + 1 \).
- \( f(f(x)) \) and \( g(g(x)) \) are also not equal, as \( x^4 \neq 4x + 3 \).

Thus, we conclude that \( f(g(x)) \neq g(f(x)) \) and \( f(f(x)) \neq g(g(x)) \).