Problem 79 - College Algebra with Modeling and Visualization

Question

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x+1)-2\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 2 & 4 & 3 & 7 & 8 & 10\end{array}$$

Step-by-Step Solution

Answer

The numerical representation of \(g(x)\) is \(g(1)=2, g(2)=1, g(3)=5, g(4)=6, g(5)=8\).

Step 1: Understanding the relationship

We are given the relationship between the functions: \(g(x) = f(x+1) - 2\). This means to find \(g(x)\), we need to take the value of \(f\) at \(x+1\) and then subtract 2 from it.

Step 2: Analyzing given data for function \(f\)

The table provides us with specific values of \(f(x)\) for different \(x\): - \(f(1) = 2\)- \(f(2) = 4\)- \(f(3) = 3\)- \(f(4) = 7\)- \(f(5) = 8\)- \(f(6) = 10\).

Step 3: Calculating \(g(1)\)

To find \(g(1)\), use \(f(2)\) because \(g(1) = f(1+1) - 2 = f(2) - 2\). Thus, \(g(1) = 4 - 2 = 2\).

Step 4: Calculating \(g(2)\)

For \(g(2)\), use \(f(3)\): \(g(2) = f(2+1) - 2 = f(3) - 2\). Thus, \(g(2) = 3 - 2 = 1\).

Step 5: Calculating \(g(3)\)

For \(g(3)\), use \(f(4)\): \(g(3) = f(3+1) - 2 = f(4) - 2\). Thus, \(g(3) = 7 - 2 = 5\).

Step 6: Calculating \(g(4)\)

For \(g(4)\), use \(f(5)\): \(g(4) = f(4+1) - 2 = f(5) - 2\). Thus, \(g(4) = 8 - 2 = 6\).

Step 7: Calculating \(g(5)\)

For \(g(5)\), use \(f(6)\): \(g(5) = f(5+1) - 2 = f(6) - 2\). Thus, \(g(5) = 10 - 2 = 8\).

Step 8: Tabulating \(g\) values

Now that we have calculated the values:- \(g(1) = 2\)- \(g(2) = 1\)- \(g(3) = 5\)- \(g(4) = 6\)- \(g(5) = 8\).

Key Concepts

Numerical Representation of Functions

When dealing with functions, a numerical representation helps us understand how a function behaves by showing input-output pairs.
For the functions in our exercise, we use numbers to represent specific points, making it easier to evaluate and manipulate the functions step by step.
For instance, for function \(f(x)\), each \(x\) value has a corresponding \(f(x)\) value:

• When \(x = 1\), \(f(x) = 2\)
• When \(x = 2\), \(f(x) = 4\)

This kind of representation gives us a clear and straightforward method to handle functions, allowing you to calculate other function values like \(g(x)\) based on transformations.

Function Tables

A function table organizes values of functions into columns and rows, making numerical data easy to read and compare.
In the provided exercise, the table for function \(f(x)\) signifies how different \(x\) inputs correspond to specific \(f(x)\) values, like:

• For \(x = 3\), \(f(x) = 3\)
• For \(x = 4\), \(f(x) = 7\)

Moreover, function tables are crucial for quickly identifying the number needed in transformations or operations, as we do when calculating \(g(x)\). By organizing data systematically, tables provide a visual cue to solve function-related problems efficiently.

Function Operations

Function operations involve manipulating functions to produce new ones.
In our exercise, this involves transforming \(f(x)\) to get \(g(x)\) using the relationship \(g(x) = f(x+1) - 2\).
To perform this:

• Locate \(f(x+1)\) from the table for each given \(x\)
• Subtract 2 to find \(g(x)\)

For example, to find \(g(1)\), we see \(f(2) = 4\) and calculate \(g(1) = 4 - 2 = 2\).
This method showcases how simple arithmetic operations can be applied systematically to functions, resulting in transformations or entirely new functions.