Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ g\left(f\left(\frac{2}{3}\right)\right) $$

Step-by-Step Solution

Answer

The value of the composition is 0.

Step 1: Evaluate Inner Function

First, we need to evaluate the inner function, which is the function \(f(x)\). Substitute \(\frac{2}{3}\) into \(f(x) = 3x\). This gives us: \[ f\left(\frac{2}{3}\right) = 3 \times \frac{2}{3} = 2 \]

Step 2: Substitute Result into Outer Function

Now, we take the result from the first step and substitute it into the outer function \(g(x)\). Since the result is \(2\), substitute \(2\) into \(g(x) = x - 2\). This gives us: \[ g(2) = 2 - 2 = 0 \]

Step 3: Write the Final Result

The final result of evaluating the composition \(g\left(f\left(\frac{2}{3}\right)\right)\) is \(0\).

Key Concepts

Inner Function Evaluation

In function composition, evaluating the inner function is the first crucial step. In this process, we focus on the function that operates directly on the initial input value. For our exercise, the inner function is \(f(x) = 3x\). To evaluate, substitute the specific value given in the problem, which in this case, is \(\frac{2}{3}\). Perform the calculation:

• Multiply 3 by \(\frac{2}{3}\). This is simple arithmetic multiplication, resulting in 2.

The logic behind solving inner functions first is akin to solving operations inside parentheses before tackling the rest of the equation. This evaluation sets the groundwork for further steps in function composition. Understanding this step helps ensure the accuracy of the overall result.

Outer Function Evaluation

After successfully evaluating the inner function, the next step is the outer function evaluation. This involves using the result from the inner function as input for the next function in the composition. In our case, the outer function is \(g(x) = x - 2\). Previously, we found that \(f\left(\frac{2}{3}\right) = 2\). Now, let's substitute this result into the outer function:

• Replace \(x\) in \(g(x)\) with 2, resulting in \(g(2) = 2 - 2\).
• Simplify this to get \(0\).

The outer function evaluation completes the composition process. Correctly handling this part confirms the flow of operations, leading us to the final result. The premise is to sequentially solve from inside out, ensuring each step uses accurate inputs from the preceding calculation.

Substitution Method

The substitution method in function composition involves strategically inserting the computed result of one function into another. This method is crucial when dealing with nested functions or when multiple functions act in sequence on a variable. In the context of our problem, once the inner function \(f(x)\) was evaluated:

• The resultant \(f\left(\frac{2}{3}\right) = 2\) is then substituted into the outer function \(g(x)\).
• It transforms the outer function into \(g(2)\), subsequently leading to a straightforward computation.

Substitution is an essential strategy for handling composite functions and ensures that each function uses data processed from previous steps. Through substitution, we maintain the continuity and integrity of calculations, guiding us smoothly from start to finish in solving composite functions effectively.