Problem 61 - College Algebra

Question

Use the function values for \(f\) and \(g\) shown in Table 3 to evaluate each expression. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 7 & 6 & 5 & 8 & 4 & 0 & 2 & 1 & 9 & 3 \\\ \hline \boldsymbol{g}(\boldsymbol{x}) & 9 & 5 & 6 & 2 & 1 & 8 & 7 & 3 & 4 & 0 \\\ \hline \end{array} $$ $$ g(f(3)) $$

Step-by-Step Solution

Answer

4

Step 1: Find f(3)

Look at the table to find the value of \( f(3) \). According to the table, when \( x = 3 \), the value of \( f(x) \) is 8. Therefore, \( f(3) = 8 \).

Step 2: Evaluate g(f(3))

Now that we know \( f(3) = 8 \), we need to find \( g(8) \). Look at the table to find the value of \( g(8) \). According to the table, when \( x = 8 \), the value of \( g(x) \) is 4. Therefore, \( g(8) = 4 \).

Step 3: Conclusion

Since we found that \( g(f(3)) = g(8) = 4 \), the final value of the expression is 4.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics where you determine the output of a function given a specific input. Think of a function like a machine. You insert a number, it performs its internal operations, and out pops a result. This process is extremely important to understand since it helps you interact with and manipulate mathematical expressions.
For instance, if you are asked to evaluate a function, such as finding the value of \(f(3)\), you are simply determining what the function outputs when 3 is the input. Checking a table or a formula gives you the corresponding output, like reading off a menu.
When working with function evaluation, keep the following points in mind:

• Identify the input value and what function you are working with.
• Locate the corresponding output using available resources, like a table or an equation.
• Remember that each function has its own set of rules for determining outputs.

This process helps break down more complex operations, such as composing two functions.

Table Lookup

A table lookup is a simple yet powerful tool for evaluating functions. It refers to using a table containing pre-determined values to find the output of a function for a specific input. This method is useful when dealing with known function values that are difficult or tedious to calculate on-the-fly.
For example, when using a table to evaluate \(f(3)\), you simply find the row or column labeled with the input, and read across to find the output. In our example, the table shows that when \(x = 3\), \(f(x) = 8\).
Advantages of using table lookup include:

• Quick and easy retrieval of function values.
• Eliminates the need for complex calculations during each function evaluation.
• Helps verify calculations by providing reference values.

However, one should ensure the table is accurately filled and updated to provide the right values. It's also crucial to navigate the table properly to avoid mixing up inputs and outputs.

Algebraic Expressions

Algebraic expressions form the backbone of many problems involving function evaluation and composition. These expressions consist of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division.
When dealing with problems such as \(g(f(x))\), algebraic expressions help in understanding and solving these nested functions systematically. You evaluate the inner function first, for example, \(f(3)\), then use that result to find \(g\) of the derived value, \(g(8)\) in this case.
Key aspects of working with algebraic expressions include:

• Understanding the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Breaking down complex expressions into smaller, more manageable steps.
• Using known values (like those obtained from table lookups) to simplify evaluations.

This process not only aids in understanding how various functions interconnect but also develops skills to dissect and solve more abstract problems involving algebra.