Problem 42 - Precalculus : Building Concepts and Connections

Question

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \(g(4)\)

Step-by-Step Solution

Answer

The value of \(g(4)\) is 3.

Step 1: Understand The Exercise

This exercise involves finding the value of the function \(g\) at \(x=4\). First, look at the table provided for function \(g\). The first row of the table represents the \(x\) values and the second row represents the corresponding \(g(x)\) values.

Step 2: Locate \(x=4\) on the Table

Locate \(x=4\) on the first row (top row) of the table of \(g\). \(x\) values are given in the first row, so this is where to look for the \(x=4\) value.

Step 3: Find The Corresponding \(g(x)\) Value

Once the \(x=4\) value is located, look directly below it to the second row. This is the \(g(x)\) value corresponding to \(x=4\).

Key Concepts

Tables of Values

When dealing with functions in mathematics, tables of values can be a valuable tool! They provide a simple way to identify specific values of a function at given points. A table of values for a function typically consists of two rows: one representing the input values (often termed as the \(x\) values) and the other representing the output values (often noted as \(f(x)\) or \(g(x)\), depending on the function).

The key idea here is that each entry in the top row directly relates to an entry beneath it in the bottom row. This structure allows us to quickly determine the output of the function at any given input value by simply scanning across the row for the desired input value. Therefore, understanding how to read tables of values is crucial when evaluating functions.

Corresponding Values

In any table of values, the concept of 'corresponding values' is vital. For a given input \(x\), the corresponding value is the output value of the function, either \(f(x)\) or \(g(x)\), that appears directly below the chosen input in the table.

- Let's use our table for function \(g\) as an example: - If you want to find the value of \(g(4)\), look for \(x=4\) in the top row. - Once you find \(x=4\), look directly below it in the second row – that value is \(g(4)\).

Learning to identify corresponding values quickly can make evaluating functions with tables much more straightforward. This understanding allows you to translate the visual information from a table into actionable numerical results.

Step by Step Solution

Understanding and following a step by step solution can greatly simplify working with functions and their tables of values. Let's break down the process, using the evaluation of \(g(4)\) as an example:

• Step 1: Understand the Exercise
  First, ensure you recognize what the problem asks for. Here, it's the value of \(g(x)\) when \(x=4\). This requires locating the entry corresponding to \(x=4\) in the table for \(g\).
• Step 2: Locate \(x=4\) on the Table
  Focus on the top row of the table, which gives you different values for \(x\). Search for \(4\) in this row.
• Step 3: Find the Corresponding \(g(x)\) Value
  Once found, look directly beneath \(x=4\) in the second row, which will show \(g(4)\). In our example, \(g(4)\) equals \(3\).

By breaking down the solution process into clear and logical steps, any function evaluation can become manageable. This approach systematically addresses how to translate table entries into specific function values.