Problem 53 - College Algebra

Question

f and g are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{cc}x & f(x) \\ \hline-1 & 1 \\ 0 & 4 \\ 1 & 5 \\ 2 & -1 \end{array}$$ $$\begin{array}{cc}x & g(x) \\ \hline-1 & 0 \\ 1 & 1 \\ 4 & 2 \\ 10 & -1 \end{array}$$ $$ f(g(1)) $$

Step-by-Step Solution

Answer

According to the provided tables, the value of \(f(g(1))\) is 5.

Step 1: Evaluating the inner function

First, we need to evaluate the inner function of our composite function. Which is, we need to find the value of \(g(1)\) by looking at the table of g(x). The table shows that the value of g at x=1 is 1.

Step 2: Evaluating the outer function

Now, we can evaluate the outer function. This means we use the value from the previous step to evaluate the main function f. In this case, we need to find the value of \(f(g(1))\) or \(f(1)\), which is also 5.

Key Concepts

Understanding Function Evaluation

Function evaluation is a fundamental concept in mathematics, especially when dealing with functions and their compositions. It involves substituting an input value into a function to find the corresponding output. In the context of the given exercise, we're looking at composite functions, where one function's output becomes the input for another function.

To evaluate a function, follow these simple steps:

• Identify the function you need to evaluate and determine the input value you must substitute into the function.
• Use this input value to find the corresponding output in a predefined table or by directly substituting it into the function's equation.
• In composite functions, start by evaluating the innermost function first and use its output as the input for the next function.

For example, if we want to evaluate the composite function \(f(g(1))\), we first find \(g(1)\), and then evaluate \(f\) at this output. It’s like a step-by-step process, ensuring that each function is correctly applied before moving on to the next.

Function Tables: A Practical Tool

Function tables are essential tools that help simplify the process of finding function values. They provide a straightforward way to view and compare input-output pairs for different functions. Let’s break down how to use them effectively:

You start by locating the input value in the left column of the table. The corresponding value in the right column is the function’s output. This method is useful for discrete functions, where calculations might otherwise require more complex operations.

In our exercise, two tables are given: one for \(f(x)\) and another for \(g(x)\). Each table lists specific pairs of inputs and outputs. For \(g(x)\), when \(x = 1\), the output is 1, from the given table information. This simple lookup process saves time and reduces the risk of errors when dealing with composite functions. By having this information organized in tables, you get to visualize the relationship between inputs and outputs clearly. You can quickly see that after evaluating \(g\), the next step is to use this result in the \(f(x)\) table for the final evaluation.

Understanding Inner and Outer Functions

When dealing with composite functions, the terms 'inner function' and 'outer function' describe the roles functions play in the overall evaluation process. Recognizing these roles is crucial for understanding how to execute function compositions.

Think of the inner function as the function that you evaluate first. This function directly uses the initial given input. In our case, \(g(x)\) is the inner function because it’s evaluated at x = 1 initially. Once you determine the output of the inner function, this output becomes the input for the outer function.

The outer function works with the result from the inner function to produce the final output of the composite function. In this scenario, \(f(x)\) is the outer function. Once \(g(1) = 1\) is known, it becomes the input for \(f\), leading to the evaluation \(f(1)\), which equals 5.

This structure mimics a nesting effect, where functions are layered, and each needs to be processed in a specific order. Understanding which function is inner and which is outer ensures you're following the correct order of operations for accurate results.