Question

Let \( f(x) = 3 + 2x \) and \( g_n(x) = (f \circ f \circ \dots \text{n times}) (x) \). If for all \( n \in \mathbb{N} \), the lines \( y = g_n(x) \) pass through a fixed point \( (a, \beta) \), then \( \alpha + \beta = ? \)

• A. \( -5 \)
• B. \( -4 \)
• C. \( -3 \)
• D. \( -6 \)

Hint:

To solve problems involving repeated application of functions, look for patterns in the outputs after applying the function multiple times. For linear functions, you can derive a general expression for \( g_n(x) \) by recognizing the structure of the function.

Answer 1

The Correct Option is D

We are given that \( f(x) = 3 + 2x \) and \( g_n(x) = (f \circ f \circ \dots \text{n times}) (x) \), and the lines \( y = g_n(x) \) pass through a fixed point \( (a, \beta) \). We need to determine the value of \( \alpha + \beta \). 
Step 1: 
First, let us calculate \( g_n(x) \). Since \( f(x) = 3 + 2x \), we apply this function \( n \) times. For \( g_1(x) = f(x) \), we have: \[ g_1(x) = 3 + 2x. \] Now, applying \( f(x) \) again, we get \( g_2(x) = f(f(x)) \): \[ g_2(x) = f(3 + 2x) = 3 + 2(3 + 2x) = 3 + 6 + 4x = 9 + 4x. \] Next, applying \( f(x) \) once more for \( g_3(x) \), we get: \[ g_3(x) = f(9 + 4x) = 3 + 2(9 + 4x) = 3 + 18 + 8x = 21 + 8x. \] Observing the pattern, we can generalize the expression for \( g_n(x) \) as: \[ g_n(x) = 3 \cdot (2^n - 1) + 2^n x. \] 
Step 2: 
We are told that the lines \( y = g_n(x) \) pass through a fixed point \( (a, \beta) \), which implies that for all \( n \), the equation \( g_n(a) = \beta \) must hold. Substituting \( x = a \) into the general expression for \( g_n(x) \), we get: \[ g_n(a) = 3 \cdot (2^n - 1) + 2^n a. \] Since this equation holds for all \( n \), we can find the value of \( a \) and \( \beta \) by setting the coefficients of \( 2^n \) equal and constant for all \( n \). From the structure of \( g_n(x) \), we observe that the only possibility for the values of \( a \) and \( \beta \) to satisfy the condition for all \( n \) is \( a = -3 \) and \( \beta = -3 \). Thus, \( \alpha + \beta = -6 \).