Problem 77 - 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 \(\mathbf{g}\). \(g(x)=f(x-2)\) $$\begin{array}{rrrrrr}x & -4 & -2 & 0 & 2 & 4 \\\f(x) & 5 & 2 & -3 & -5 & -9\end{array}$$

Step-by-Step Solution

Answer

g(0)=2, g(2)=-3, g(4)=-5; g(-4) and g(-2) cannot be determined.

Step 1: Understand the Relationship

We are given the function equation \( g(x) = f(x-2) \). This means that to find \( g(x) \), we must use the values of \( f(x) \) by adjusting \( x \) by adding 2, i.e., substituting \( x-2 \) into \( f \).

Step 2: Calculate g(x) for Each x Value

To compute \( g(x) \) at specific points, we simply replace the points in the definition of \( g(x) = f(x-2) \). For example, to find \( g(-4) \), calculate \( f(-4-2) = f(-6) \), but since \( f(-6) \) is not provided, we can't determine \( g(-4) \) from the given data.

Step 3: Practical Application

Evaluate \( g(x) \) using available \( f(x) \) values:- \( g(0) = f(0-2) = f(-2) = 2 \)- \( g(2) = f(2-2) = f(0) = -3 \)- \( g(4) = f(4-2) = f(2) = -5 \).Unavailable \( f \) values imply that \( g(-4) \) and \( g(-2) \) cannot be determined from the given data.

Step 4: Construct the Numerical Representation of g(x)

Based on available calculations, the function \( g(x) \) can be represented numerically as follows:- \( g(0) = 2 \)- \( g(2) = -3 \)- \( g(4) = -5 \)This is based on the corresponding \( f(x) \) values provided for the respective shifted \( x \, values \).

Key Concepts

Numerical Representation

Numerical representation refers to expressing a function in terms of specific numerical inputs and outputs. To understand this better, think of a function table as a map. The "x" values are the locations you want to visit, and "f(x)" values are the scenes at each location. In our exercise, we are given a table of values for function \(f\) at different locations:

• \((-4, 5)\)
• \((-2, 2)\)
• \((0, -3)\)
• \((2, -5)\)
• \((4, -9)\)

These pairs show how each x-value corresponds to an output. This representation is crucial for understanding the behavior of function \(f\). It's a simple way to see particular points on a graph, giving us a quick snapshot of the function's behavior without needing a full graph.

Function Evaluation

Function evaluation is the process of determining the output of a function given an input. It helps in understanding how two functions relate to each other. In our exercise, the function \(g(x)\) is defined in terms of \(f(x)\) as \(g(x) = f(x-2)\). Each x-value in \(g\) is calculated by "shifting" the input of \(f\) by 2 to the right.
For instance, to evaluate \(g(0)\):

• Calculate \(f(0-2)\), meaning you find \(f(-2)\).
• From the table, \(f(-2) = 2\), so \(g(0) = 2\).

By repeating this process, you can find outputs for different inputs in \(g(x)\). If the required input for \(f\) is not listed in the table, like \(f(-6)\) for \(g(-4)\), the function \(g\) cannot be evaluated for those specific points based on the given data.

Shifted Functions

Shifted functions are transformations where the entire graph of a function moves in a specific direction. In the context of our exercise, \(g(x) = f(x-2)\) illustrates a "horizontal shift." This means you take the graph of \(f(x)\) and shift it 2 units to the right to get \(g(x)\).
Here's how horizontal shifts work:

• A positive constant inside the function \(f(x-c)\) shifts the graph to the right by "c" units.
• So, \(g(x) = f(x-2)\) means a rightward shift of 2 units.
• If it were \(f(x+2)\), the graph would shift leftward by 2 units.

This shifting process doesn't alter the shape of the function graph; it only changes its position on the x-axis. Consequently, recognizing shifts is essential in predicting function behaviors without re-plotting complex graphs.