Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Solve \(f(x)=3\).

Step-by-Step Solution

Answer

\(x = 7\) makes \(f(x) = 3\).

Step 1: Understand the Problem

We need to find the value of \(x\) that makes \(f(x) = 3\). This means we need to look for where the function \(f\) outputs the value 3.

Step 2: Analyze the Table

Examine the given table, where \(x\) values are listed in the first row, and corresponding \(f(x)\) values are listed in the second row. Locate the position where \(f(x) = 3\).

Step 3: Locate \(f(x) = 3\)

Look at each column in the table: When \(x = 7\), \(f(x) = 3\). Thus, \(f(7) = 3\).

Step 4: Verify the Solution

Revisit the function value for \(x = 7\) and confirm that \(f(7)\) equals 3 as per the table, ensuring no mistakes in reading the table.

Key Concepts

Discrete Mathematics

Discrete mathematics is a fascinating area of mathematics that deals with distinct and separate values. It is crucial for understanding how to manage datasets that are not continuous but rather consist of isolated points—in our case, these points are represented as pairs of numbers in a table. Discrete mathematics is often employed in computer science, cryptography, and information theory because it helps us analyze, among other things, functions represented by individual data points rather than continuous curves.
When solving problems like the one presented with a function evaluation, we make discrete choices about values. Each choice is a unique, standalone decision, such as identifying a specific "x" value that corresponds to a given "f(x)" value in a table. This framework allows us to focus on direct and exact relationships without worrying about interpolating between data points.

Table Analysis

Analyzing tables is a fundamental skill in mathematics that helps visualize relationships and dependencies between variables. In our example, a table is used to represent a discrete function, where a set of distinct "x" values has corresponding "f(x)" outputs. Table analysis involves interpreting this structure to find desired information—like determining where the function reaches a specific value.
Here’s how you can effectively approach table analysis:

• Understand the axis: Recognize that the table’s rows represent independent and dependent variables, often corresponding to "x" and "f(x)", respectively.
• Locate target values: Scan through the relevant row or column to identify positions that match the criteria you are solving for. In this instance, looking for where "f(x) = 3" was critical.
• Verify findings: Always double-check to ensure the right correspondence between variables—this confirms the solution's accuracy.

Table analysis simplifies complex problems by breaking them down into manageable pieces. This enables a systematic approach to solving algebraic and discrete data problems.

Algebraic Problem Solving

Algebraic problem solving is a versatile tool in both applied and theoretical mathematics, involving the use of equations and functions to describe relationships between quantities. Functions are core to algebra because they create connections between sets of inputs and outputs.
In the problem at hand, the function is expressed as a table, defining the output for each possible input directly. These kinds of problems often require identifying specific inputs that will produce desired outputs, akin to solving the equation "f(x) = 3". This process can be broken down as follows:

• Define the function: Know what each input 'x' will produce as output 'f(x)' and understand this is each row or column in the table.
• Set the equation: The problem provided is equivalent to setting an equation where the output is sought (here, 3).
• Relate and solve: Match the desired outcome with the input value that provides it, checking against the table or using algebraic manipulation if needed.

Algebraic problem solving in this discrete context is about mapping out exact answers rather than estimating solutions, which aligns with the principles of discrete mathematics.