Problem 80 - 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 \(\mathbf{g}\). \(g(x)=f(x-3)+5\) $$\begin{array}{rrrrrr}x & -3 & 0 & 3 & 6 & 9 \\\\\hline f(x) & 3 & 8 & 15 & 27 & 31\end{array}$$

Step-by-Step Solution

Answer

\(g(x)\) values are 8, 13, 20, 32 for \(x\) of 0, 3, 6, 9 respectively.

Step 1: Understand the given equation

The problem provides the equation \( g(x) = f(x-3) + 5 \). This equation tells us that the function \( g \) at each point \( x \) is equal to the function \( f \) calculated at \( x-3 \), plus 5.

Step 2: Identify required input for f

For a given \( g(x) \), we need the value of \( f(x-3) \), which means we need the input \( x-3 \) that corresponds to \( x \) in the table for \( f \). Calculate \( x-3 \) for each \( x \).

Step 3: Calculate x-3 for each x in g(x)

For each \( x \) value of \( -3, 0, 3, 6, 9 \), calculate \( x-3 \) as follows: - If \( x = -3 \), then \( x-3 = -6 \) (not available in the table)- If \( x = 0 \), then \( x-3 = -3 \)- If \( x = 3 \), then \( x-3 = 0 \)- If \( x = 6 \), then \( x-3 = 3 \)- If \( x = 9 \), then \( x-3 = 6 \)

Step 4: Find corresponding f(x-3) values

Using the adjusted \( x-3 \) values, find the corresponding \( f(x-3) \):- For \( x = 0 \), \( f(-3) = 3 \)- For \( x = 3 \), \( f(0) = 8 \)- For \( x = 6 \), \( f(3) = 15 \)- For \( x = 9 \), \( f(6) = 27 \)- \( x = -3 \) yields \( f(-6) \) which is not given (skip).

Step 5: Apply the function g(x)

Apply the equation \( g(x) = f(x-3) + 5 \) by adding 5 to each \( f(x-3) \) value calculated:- For \( x = 0 \), \( g(0) = 3 + 5 = 8 \)- For \( x = 3 \), \( g(3) = 8 + 5 = 13 \)- For \( x = 6 \), \( g(6) = 15 + 5 = 20 \)- For \( x = 9 \), \( g(9) = 27 + 5 = 32 \)- \( x = -3 \) cannot be calculated as it's out of the table's range.

Step 6: Create the numerical representation of g(x)

Record the calculated \( g(x) \) values in a new table for available inputs:\[\begin{array}{cc}x & g(x) \0 & 8 \3 & 13 \6 & 20 \9 & 32\end{array}\]

Key Concepts

Numerical Representation

Numerical representation in mathematics is a way to express relationships between variables in a table format using numbers. This is especially handy in understanding the transformation of functions. In this problem, we started with data for the function \( f(x) \), presented as a set of corresponding \( x \) and \( f(x) \) values.

The goal was to create a numerical representation for \( g(x) \), based on a transformation of \( f(x) \). Given the equation \( g(x) = f(x-3) + 5 \), each \( g(x) \) value is calculated by modifying \( x \) to \( x-3 \), finding the corresponding \( f(x-3) \) value, and adding 5. For instance:

• For \( x = 0 \), \( x-3 = -3 \), and \( f(-3) = 3 \). So \( g(0) = 3 + 5 = 8 \).
• For \( x = 3 \), \( x-3 = 0 \), and \( f(0) = 8 \). Thus, \( g(3) = 8 + 5 = 13 \).
• And so on for other \( x \) values that are within the provided table range.

This systematic approach ensures each step of the transformation is clearly understood, through numerical data.

Function Notation

Function notation is a concise way to represent and analyze functions. It uses symbols to describe the relationship between inputs, also called independent variables, and outputs, or dependent variables. In our exercise, we have two functions: \( f \) and \( g \).

Using function notation, we express the transformation of \( f \) to \( g \) as \( g(x) = f(x-3) + 5 \). This means that for every \( x \), \( g(x) \) is derived by first computing \( f(x-3) \)--a shifted version of the original input--and then adding 5 to the result. This notation succinctly captures the transformation:

• The "\( x-3 \)" inside \( f \) implies a shift in the input space.
• Adding 5 indicates a vertical shift or translation in the output.

Function notation helps visualize how inputs are manipulated and how outputs are obtained, aiding in clearer understanding of transformations.

Function Table

Function tables are structured ways to display how transformations affect inputs and outputs. By listing \( x \) values alongside their corresponding \( f(x) \) and \( g(x) \) values, tables help reveal patterns and relationships within a function or transformation.

In this exercise, the function table for \( f(x) \) showed how values progressed from \(-3\) to \(9\). To find \( g(x) \), adjustments were made using the \( x-3 \) transformation, and the results were documented in a new table:

Considerations for building a function table:

• Identify the range of \( x \) values and compute the transformed \( x-3 \).
• Retrieve \( f(x-3) \) values for all valid entries (when \( x-3 \) corresponds to a value in the \( f(x) \) table).
• Add 5 to each \( f(x-3) \) to get \( g(x) \), compiling the results systematically.

Using a function table this way clarifies the impact of transformations like shifts and translations, making complex equations more tangible and concrete.