Problem 54 - Algebra 2

Question

If \(f(0)=4\) and \(f(x+1)=3 f(x)-2,\) find \(f(4)\)

Step-by-Step Solution

Answer

\(f(4) = 244\).

Step 1: Understand the Problem

We are given an initial condition for a function, say \(f(0) = 4\), and a recursive formula \(f(x+1) = 3f(x) - 2\). Our task is to use these to find \(f(4)\).

Step 2: Calculate f(1) using the Recursive Formula

Substitute \(x = 0\) into the recursive equation. We have \(f(1) = 3f(0) - 2\). Since \(f(0) = 4\), it follows that \(f(1) = 3 \times 4 - 2 = 12 - 2 = 10\).

Step 3: Calculate f(2) using the Recursive Formula

Substitute \(x = 1\) into the recursive equation. We have \(f(2) = 3f(1) - 2\). Using \(f(1) = 10\), it follows that \(f(2) = 3 \times 10 - 2 = 30 - 2 = 28\).

Step 4: Calculate f(3) using the Recursive Formula

Substitute \(x = 2\) into the recursive equation. We have \(f(3) = 3f(2) - 2\). Using \(f(2) = 28\), it follows that \(f(3) = 3 \times 28 - 2 = 84 - 2 = 82\).

Step 5: Calculate f(4) using the Recursive Formula

Substitute \(x = 3\) into the recursive equation. We have \(f(4) = 3f(3) - 2\). Using \(f(3) = 82\), it follows that \(f(4) = 3 \times 82 - 2 = 246 - 2 = 244\).

Step 6: Final Answer

The value of \(f(4)\) is 244.

Key Concepts

Initial Condition

An initial condition is a crucial starting point in solving problems involving recursive functions. In this exercise, the initial condition is given as \( f(0) = 4 \). This simply means that when \( x = 0 \), the function \( f(x) \) has a value of 4.

Initial conditions help to provide the foundation required to determine all subsequent values in the sequence. Without it, you wouldn't be able to use the recursive formula effectively.

Think of the initial condition as the "launch pad" for the calculations; it gives us a definite known value from which we can apply further rules, like our recursive formula, to generate a sequence of values.

Recursive Formula

The recursive formula acts like a blueprint for constructing the sequence of values in a function based on previous values. In this case, the recursive formula is \( f(x+1) = 3f(x) - 2 \).

This formula gives a method to calculate each next value in the sequence by using the previous one. Here's how it works:

• For any \( x \), it tells us how to find \( f(x+1) \) using the value of \( f(x) \).


• The formula \( 3f(x) - 2 \) signifies taking the current value of \( f(x) \), multiplying it by 3, and then subtracting 2 to get the next value.

Recursive formulas are powerful in finding terms in a sequence without needing to know them all explicitly,—just the previous one, which makes them much more efficient than other methods like direct computation.

Step by Step Calculation

Using the initial condition and recursive formula, we can find \( f(4) \) by following a simple pathway of calculations. Here’s how we do this step by step:

1. **Start with the Initial Condition:** We know \( f(0) = 4 \). This is our beginning point for finding other values.

2. **Calculate \( f(1) \):** Using the recursive formula, substitute \( x = 0 \) to get \( f(1) = 3 \times 4 - 2 = 10 \).

3. **Calculate \( f(2) \):** Next, substitute \( x = 1 \) into the recursive formula to find \( f(2) = 3 \times 10 - 2 = 28 \).

4. **Calculate \( f(3) \):** Now, use \( x = 2 \) which gives \( f(3) = 3 \times 28 - 2 = 82 \).

5. **Finally, Calculate \( f(4) \):** Substitute \( x = 3 \) to determine \( f(4) = 3 \times 82 - 2 = 244 \).

Each step uses the result of the previous step to determine the next, making it a clear and logical process to arrive at the desired function value.