Problem 75 - 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 \(\mathbf{g}\). \(g(x)=f(x)+7\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 5 & 1 & 6 & 2 & 7 & 9\end{array}$$

Step-by-Step Solution

Answer

The new table for \( g(x) \) is: \( g(x) = [12, 8, 13, 9, 14, 16] \).

Step 1: Understand the relationship

We begin by understanding the relationship between the functions, which is provided as \( g(x) = f(x) + 7 \). This means that each output value of \( g \) is formed by adding 7 to the corresponding output value of \( f \).

Step 2: Process each pair of inputs and outputs

We will use the table of \( f(x) \) values to compute \( g(x) \) at each input \( x \):- For \( x = 1 \), \( f(x) = 5 \), so \( g(1) = 5 + 7 = 12 \).- For \( x = 2 \), \( f(x) = 1 \), so \( g(2) = 1 + 7 = 8 \).- For \( x = 3 \), \( f(x) = 6 \), so \( g(3) = 6 + 7 = 13 \).- For \( x = 4 \), \( f(x) = 2 \), so \( g(4) = 2 + 7 = 9 \).- For \( x = 5 \), \( f(x) = 7 \), so \( g(5) = 7 + 7 = 14 \).- For \( x = 6 \), \( f(x) = 9 \), so \( g(6) = 9 + 7 = 16 \).

Step 3: Create the numerical representation of g

Now we compile our calculated values from the previous step into a table to represent \( g \):\[\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \hline g(x) & 12 & 8 & 13 & 9 & 14 & 16\end{array}\]

Key Concepts

Numerical Representation

In algebra, the concept of numerical representation involves expressing a function using numbers. This means that for any given function, we can represent its rules and outcomes in numerical form, usually through a table. Each row in the table often includes specific values of the input variable, commonly denoted by \(x\), and the corresponding output from the function.When we approach problems like the one in the example, we're given a function \(f(x)\) that provides specific outputs for each input. By following the function's definitive rule, we apply this to every corresponding value from its list. The result is a clear, straightforward table that reflects the function's nature through numbers.Using a numerical representation not only helps in understanding the function's behavior but also prepares us to manipulate and transform the function in various ways. This step is foundational in learning about functions in algebra.

Function Transformation

Function transformation is a fundamental concept in algebra. It involves changing a function in various ways to explore different properties or to fit certain criteria. In our exercise, we are asked to transform the function \(f(x)\) into \(g(x)\). The transformation here is straightforward because the relationship is given by \(g(x) = f(x) + 7\).This means that every output of \(f(x)\) is increased or 'shifted' by 7 to produce \(g(x)\). Function transformations frequently involve operations like addition, subtraction, multiplication, and division over the function's output. Indeed, our example employs a vertical shift upwards by adding a constant, which is one of the most common transformations in algebra.Understanding function transformations allows us to predict how changes affect the graph or table of a function. It is key to refining mathematical models and solving complex algebraic problems.

Function Table Values

The function table is a crucial tool for visualizing and calculating values produced by functions in algebra. It rows each pair of input-output values that meet the function's rule. In the provided example, we use the function table method to determine the new values of \(g(x)\).Here's how the process unfolds:

• We start with a table for \(f(x)\) that lists each \(x\) and its respective \(f(x)\) value.
• To find \(g(x)\), which is \(f(x) + 7\), we simply add 7 to every value of \(f(x)\).
• The resulting table reveals the new values of \(g(x)\) for each \(x\).

By using function table values, you can easily track how changes in the function's rule affect its outcomes. It's a practical way of testing, verifying, and comparing the function's output, especially when handling transformations.