Problem 11 - Applied Calculus

Question

Use Table 1.37 to find: (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(4))\) (d) \(g(f(4))\) (e) \(f(g(6))\) (f) \(g(f(6))\) $$\begin{array}{c|c|c|c|c|c|c}\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 5 & 4 & 3 & 3 & 4 & 5 \\\\\hline g(x) & 6 & 5 & 4 & 3 & 2 & 1 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) 5, (b) 2, (c) 3, (d) 4, (e) 5, (f) 2.

Step 1: Evaluate f(g(1))

To find \( f(g(1)) \), we need to first find \( g(1) \). From the table, we see that \( g(1) = 6 \). Next, we need to find \( f(6) \). According to the table, \( f(6) = 5 \). Therefore, \( f(g(1)) = 5 \).

Step 2: Evaluate g(f(1))

To find \( g(f(1)) \), we first determine \( f(1) \). From the table, \( f(1) = 5 \). Next, we find \( g(5) \). From the table, \( g(5) = 2 \). Therefore, \( g(f(1)) = 2 \).

Step 3: Evaluate f(g(4))

To calculate \( f(g(4)) \), we initially find \( g(4) \). According to the table, \( g(4) = 3 \). Then, we find \( f(3) \). From the table, \( f(3) = 3 \). Therefore, \( f(g(4)) = 3 \).

Step 4: Evaluate g(f(4))

To evaluate \( g(f(4)) \), we start with finding \( f(4) \). From the table, \( f(4) = 3 \). Then, we find \( g(3) \). According to the table, \( g(3) = 4 \). Therefore, \( g(f(4)) = 4 \).

Step 5: Evaluate f(g(6))

To find \( f(g(6)) \), we first obtain \( g(6) \). From the table, \( g(6) = 1 \). Next, we determine \( f(1) \). According to the table, \( f(1) = 5 \). Therefore, \( f(g(6)) = 5 \).

Step 6: Evaluate g(f(6))

To calculate \( g(f(6)) \), we start with \( f(6) \). From the table, \( f(6) = 5 \). We next find \( g(5) \). From the table, \( g(5) = 2 \). Therefore, \( g(f(6)) = 2 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves determining the output of a function for a specific input value. When dealing with composite functions such as \( f(g(x)) \), we perform this evaluation in two stages. First, we substitute the input \( x \) into the inner function (in this case, \( g(x) \)), resulting in an initial output. This output then becomes the input for the outer function \( f(x) \).
By following this straightforward approach, we can systematically solve each part of a composite function without confusion. Evaluating precisely ensures that we understand how inputs transform through each stage of function composition. It helps us grasp the overall behavior and relationship between functions in mathematical contexts.

Table Lookup

Tables are useful tools in mathematics for listing the values of functions at given points, making table lookup a straightforward process. In exercises involving composite functions, we often use tables to easily find needed values for intermediate steps.
For example, to evaluate \( f(g(1)) \), locate \( g(1) \) in the table and then find the corresponding value for \( f \) at that result. This process is repeated for each sub-computation within composite functions.

• Find the initial output using the table.
• Use this output to determine the next value, also utilizing the table.

This method is especially user-friendly as it minimizes errors and reinforces a clear understanding of how function inputs and outputs interrelate. Remember that reading tables accurately is a skill that becomes polished with practice.

Step-by-Step Calculation

Breaking down complex problems into smaller, manageable steps increases accuracy and comprehension. This approach, known as step-by-step calculation, is pivotal when working with composite functions.
When evaluating problems such as \( f(g(4)) \), for instance, determine \( g(4) \) first, using the corresponding table values. Follow by computing \( f(g(4)) \), ensuring a clear path from start to finish.
Step-by-step calculations:

• Work systematically from inner to outer functions.
• Check table values twice to avoid mistakes.
• Write down intermediary results to maintain clarity.

Using this structured method not only aids in finding the correct solution but also builds a stronger foundation for more complex mathematical problems you will encounter later.

Applied Calculus

Applied calculus uses the principles of calculus in practical scenarios, such as determining changes and behaviors in functions. When dealing with composite functions, this aspect of calculus becomes apparent, particularly when interpreting the transformation of functions and how values impact one another.
The exercises presented reveal how functions can be nested, and their inputs processed through cascading operations. Understanding how to manipulate these nested functions equips students to handle real-world problems where such mathematical strategies are necessary.
For instance, using table data to discover results of composite functions simulates thinking in scenarios where raw data informs model predictions. By mastering these exercises, you learn not only how to compute but also understand how these computations can be applied in broader, practical contexts in the field of applied calculus.