Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{g}(\mathrm{f}(4)) $$

Step-by-Step Solution

Answer

The result of \(g(f(4))\) is 10.

Step 1: Evaluate the inner function f(x)

The inner function is \(f(x) = 3x\). We need to evaluate the composition \(\mathrm{g} (\mathrm{f} (4))\). Substitute \(x = 4\) into \(f(x)\) to get \(f(4) = 3 \times 4 = 12\).

Step 2: Evaluate the outer function g(x)

Now that we know \(f(4) = 12\), substitute this result into the outer function \(g(x) = x - 2\). Therefore, \(g(12) = 12 - 2 = 10\).

Key Concepts

Understanding the Inner Function

In mathematics, particularly when dealing with functions, the term **"inner function"** refers to the function that is evaluated first in a composition of functions. Imagine a situation where you have multiple functions nested within one another. In our exercise, the inner function is the one that serves as the initial stepping-stone in reaching our final value.

The inner function is written as \(f(x) = 3x\). To solve the problem, we start by evaluating this inner function. In our specific case, we need to find out what \(f(4)\) equals. This is simple:

• Substitute \(x = 4\) into \(f(x)\).
• The computation becomes \(f(4) = 3 \times 4 = 12\).

This result is crucial as it forms the foundation for evaluating the outer function. By effectively solving the inner function, we accomplish the crucial first step towards solving the entire function composition.

Decoding the Outer Function

Once you've determined the result from the inner function, your next task is to address the **"outer function"**. The outer function is what takes the outcome of the inner function to give you the final answer in a composition.

In our exercise, the outer function is \(g(x) = x - 2\). Now, remember that our inner function gave us \(f(4) = 12\). This value now becomes the input for the outer function. Here's a clear approach:

• Substitute \(f(4)\), which is 12, into \(g(x)\).
• The equation becomes \(g(12) = 12 - 2 = 10\).

Solving the outer function is usually the final step. By effectively substituting and computing, you reach the composition result. It's essential to always use the output from the inner function as the input for the outer function.

Substitute Values and Solve

Substitution is central to handling function compositions. When you hear "substitute values," think of it as inserting a specific number into a function instead of the variable, which allows you to evaluate the function’s result.

From the exercise, substitution happens twice:

• First, you substitute \(x = 4\) into the inner function, \(f(x)\), leading to \(f(4) = 12\).
• Then, you use the result \(f(4) = 12\) and substitute it into the outer function, \(g(x)\), to find \(g(12) = 10\).

By following these substitutions in sequence, you can break down complex compositions into manageable steps. Substitution is particularly helpful because it provides clarity and precision in evaluating functions. Never rush this step. Make sure your values are accurate, as precision affects the final result.