Let `g:R to R` be given by `g (x) =3+ 4x` if `g ^(n)(x) =` gogogo……og (x) n times. Then inverse of `g ^(n)(x)` is equal to :

Options

A. `(x+1-4 ^(n)). 4 ^(-n)`
B. `(x-1+4 ^(n)) 4 ^(-n)`
C. `(x+1+4^(n))4 ^(-n)`
D. None of these

Answer

To find the inverse of the function \( g^{(n)}(x) \) where \( g(x) = 3 + 4x \), we will follow these steps:

### Step 1: Understand the Function
The function \( g(x) = 3 + 4x \) is a linear function. We need to find \( g^{(n)}(x) \), which means applying the function \( g \) to itself \( n \) times.

### Step 2: Calculate \( g^{(1)}(x) \)
For \( n = 1 \):
\[ g^{(1)}(x) = g(x) = 3 + 4x \]

### Step 3: Calculate \( g^{(2)}(x) \)
For \( n = 2 \):
\[ g^{(2)}(x) = g(g(x)) = g(3 + 4x) \]
Substituting \( 3 + 4x \) into \( g(x) \):
\[ g(3 + 4x) = 3 + 4(3 + 4x) = 3 + 12 + 16x = 15 + 16x \]

### Step 4: Calculate \( g^{(3)}(x) \)
For \( n = 3 \):
\[ g^{(3)}(x) = g(g^{(2)}(x)) = g(15 + 16x) \]
Substituting \( 15 + 16x \) into \( g(x) \):
\[ g(15 + 16x) = 3 + 4(15 + 16x) = 3 + 60 + 64x = 63 + 64x \]

### Step 5: Identify the Pattern
From the calculations:
- \( g^{(1)}(x) = 3 + 4x \)
- \( g^{(2)}(x) = 15 + 16x \)
- \( g^{(3)}(x) = 63 + 64x \)

We can see a pattern forming:
\[ g^{(n)}(x) = 4^n - 1 + 4^n x \]

### Step 6: Generalize the Function
The general form for \( g^{(n)}(x) \) can be expressed as:
\[ g^{(n)}(x) = (4^n - 1) + 4^n x \]

### Step 7: Find the Inverse
To find the inverse \( g^{(n)^{-1}}(y) \), we set:
\[ y = g^{(n)}(x) = (4^n - 1) + 4^n x \]
Rearranging for \( x \):
\[ y - (4^n - 1) = 4^n x \]
\[ x = \frac{y - (4^n - 1)}{4^n} \]

### Step 8: Write the Inverse Function
Thus, the inverse function is:
\[ g^{(n)^{-1}}(y) = \frac{y - (4^n - 1)}{4^n} \]

### Summary
The inverse of \( g^{(n)}(x) \) is:
\[ g^{(n)^{-1}}(y) = \frac{y + 1 - 4^n}{4^n} \]