Problem 81 - 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 \(\mathbf{g}\). \(g(x)=f(-x)+1\) $$ \begin{array}{rrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 11 & 8 & 5 & 2 & -1 \end{array} $$

Step-by-Step Solution

Answer

The table for \(g(x)\) is \((x, g(x)): (-2, 0), (-1, 3), (0, 6), (1, 9), (2, 12)\).

Step 1: Analyze the Given Functions and Values

The function \(g(x)\) is defined as \(g(x) = f(-x) + 1\). We have a table of values for \(f(x)\). For each \(x\) in the domain of \(g\), we need to calculate \(f(-x)\) first, and then add 1 to get \(g(x)\).

Step 2: Determine f(-x) for Each x

For each value of \(x\) in the table, compute \(f(-x)\):- If \(x = -2\), \(-x = 2\), \(f(-x) = f(2) = -1\).- If \(x = -1\), \(-x = 1\), \(f(-x) = f(1) = 2\).- If \(x = 0\), \(-x = 0\), \(f(-x) = f(0) = 5\).- If \(x = 1\), \(-x = -1\), \(f(-x) = f(-1) = 8\).- If \(x = 2\), \(-x = -2\), \(f(-x) = f(-2) = 11\).

Step 3: Calculate g(x) Using g(x) = f(-x) + 1

Now compute \(g(x)\) by adding 1 to each \(f(-x)\):- \(g(-2) = f(2) + 1 = -1 + 1 = 0\).- \(g(-1) = f(1) + 1 = 2 + 1 = 3\).- \(g(0) = f(0) + 1 = 5 + 1 = 6\).- \(g(1) = f(-1) + 1 = 8 + 1 = 9\).- \(g(2) = f(-2) + 1 = 11 + 1 = 12\).

Step 4: Create the Table for g(x)

The values of \(g(x)\) based on our calculations are:\[\begin{array}{rrrrrr} x & -2 & -1 & 0 & 1 & 2 \ g(x) & 0 & 3 & 6 & 9 & 12\end{array}\]

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematics concisely. It uses symbols to relate an input to its corresponding output. When we say \(f(x)\), the function \(f\) is applied to the input \(x\), giving us an output. This notation is useful as it allows us to easily express transformations and operations on functions. In the given exercise, function \(g(x)\) is expressed in terms of another function \(f(x)\). By writing \(g(x) = f(-x) + 1\), we're explaining how \(g\) can be derived from \(f\). This involves reflecting \(f(x)\) across the y-axis by using \(-x\) and then shifting the result upward by 1 unit. Function notation helps us clearly define such transformations.

Numerical Representation

Numerical representation involves expressing functions as a set of ordered pairs \((x, f(x))\). In this context, it means using a table to display specific values of \(x\) and their corresponding outputs from the function \(f\) or \(g\). This exercise provides a numeric table for \(f(x)\) and requires us to create one for \(g(x)\).
The table method makes complex operations more manageable by focusing on individual values.

• We start with the values of \(x\) from \(-2\) to \(2\) for \(f(x)\).
• Then, calculate \(f(-x)\) by substituting each \(x\) with \(-x\).
• Add 1 to each \(f(-x)\) to find the corresponding \(g(x)\).

This step-by-step approach converts abstract algebraic operations into tangible numbers.

Function Evaluation

Function evaluation is the process of finding the output value of a function for a specific input. In the exercise, evaluating \(g(x)\) involves multiple steps as each \(g(x)\) relies on a corresponding \(f(-x)\). First, identify \(-x\) for each \(x\) and find the corresponding \(f(-x)\) from the given table.
To evaluate \(g(x)\):

• Take each \(x\) in the range and find \(-x\).
• Look up the function \(f\) using these \(-x\) values.
• Add 1 to each result to compute \(g(x) = f(-x) + 1\).

By systematically applying this evaluation process, you construct the new set of outputs for \(g(x)\). This clear breakdown of function evaluation ensures you fully comprehend how adjustments in \(x\) affect the result in the context of transformations.