Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-10\) , evaluate each composition for the given values if \(f(x)=3 x\) and \(g(x)=x-2\) $$ \mathrm{f}(\mathrm{f}(5)) $$

Step-by-Step Solution

Answer

The value of \( f(f(5)) \) is 45.

Step 1: Evaluate the Inner Function

First, we will evaluate the inner function of the composition, which is \( f(5) \). The function given is \( f(x) = 3x \), so we substitute \( x = 5 \) into this function:\[ f(5) = 3 imes 5 = 15 \]

Step 2: Evaluate the Outer Function with the Result

Now, we take the result from the inner function evaluation and use it as the input for the outer function. We need to evaluate \( f(15) \) using the same function \( f(x) = 3x \):\[ f(15) = 3 imes 15 = 45 \]

Key Concepts

Understanding the Inner Function

When dealing with function compositions, the first step is understanding the inner function. The inner function is the function you evaluate first when performing a composition. In the given exercise, the composition is expressed as \( f(f(5)) \). Here, the innermost function is \( f(5) \), which means we need to calculate this part of the expression first.

In this problem, the inner function is defined as \( f(x) = 3x \). To evaluate \( f(5) \), substitute 5 for \( x \) in the function. This results in \( f(5) = 3 \times 5 = 15 \).

• Always identify the innermost function in nested compositions.
• Substitute the given value to the inner function first.
• The output becomes the input for the subsequent function in the composition.

Recognizing and correctly evaluating the inner function is crucial in ensuring that the function composition is correct.

Defining the Outer Function

The outer function in a composition takes the result from the inner function and utilizes it for further evaluation. In our example problem \( f(f(5)) \), after computing \( f(5) = 15 \), this result is then used in the outer function. The outer function is the same as the original function \( f(x) = 3x \). Now the variable \( x \) is replaced with 15, the outcome from our inner function.

We therefore evaluate \( f(15) = 3 \times 15 = 45 \). The outer function can be considered as the final step in the composition evaluation, giving us the final value.

• The outer function uses inputs derived from inner function results.
• Think of it as an additional layer built around the inner function.
• The correctness of the final result heavily depends on correct evaluation of this function.

Understanding the role of the outer function aids in piecing together the full picture of a function composition.

Performing Function Evaluation

Function evaluation is the process of substituting a specific value or another functional result into a given function to find corresponding output. In the context of function composition, this involves evaluating each function at various stages.

Considerations for Proper Function Evaluation:

• Identify and focus on each function separately by dealing with the innermost function first.
• The interim results should be treated as inputs for subsequent functions.
• Ensure correct arithmetic operations are applied consistently.

The solution process for \( f(f(5)) \) demonstrates these fundamentals of function evaluation. Starting with the innermost function, \( f(5) \), yields 15, followed by using 15 as the input for the next function stage to get 45.
By mastering function evaluation through regular practice, understanding and computing such compositions becomes second nature.