Problem 12 - Applied Calculus

Question

Use Table 1.38 to find: (a) \(f(g(0))\) (b) \(f(g(1))\) (c) \(\quad f(g(2))\) (d) \(g(f(2))\) (e) \(g(f(3))\) $$\begin{array}{c|c|c|c|c|c|c}\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline f(x) & 10 & 6 & 3 & 4 & 7 & 11 \\\\\hline g(x) & 2 & 3 & 5 & 8 & 12 & 15 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) 3, (b) 4, (c) 11, (d) 8, (e) 12.

Step 1: Calculate f(g(0))

To find \( f(g(0)) \), first determine \( g(0) \) using the table given for \( g(x) \). We see that \( g(0) = 2 \). Next, use this result to find \( f(2) \) using the table for \( f(x) \). Looking at the table, \( f(2) = 3 \). Therefore, \( f(g(0)) = 3 \).

Step 2: Calculate f(g(1))

To determine \( f(g(1)) \), first find \( g(1) \) from the table, which gives us \( g(1) = 3 \). Next, look up \( f(3) \) using the table for \( f(x) \). The table shows \( f(3) = 4 \). Thus, \( f(g(1)) = 4 \).

Step 3: Calculate f(g(2))

First, find \( g(2) \) from the table, which is \( g(2) = 5 \). Next, look up \( f(5) \) using the value from the \( f(x) \) table. According to the table, \( f(5) = 11 \). Therefore, \( f(g(2)) = 11 \).

Step 4: Calculate g(f(2))

First, find \( f(2) \) using the table, which gives \( f(2) = 3 \). Next, find \( g(3) \) using the \( g(x) \) table, where \( g(3) = 8 \). Thus, \( g(f(2)) = 8 \).

Step 5: Calculate g(f(3))

First, find \( f(3) \) from the table, which is \( f(3) = 4 \). Next, use this result to find \( g(4) \) from the \( g(x) \) table, which shows \( g(4) = 12 \). Therefore, \( g(f(3)) = 12 \).

Key Concepts

Function Evaluation

Function evaluation refers to the process of finding the value of a function for a given input. When dealing with function composition, it often involves evaluating one function first and then using that result as the input for another function.

Let's consider an example from the given exercise: the problem of evaluating \( f(g(0)) \). Here, we need to evaluate the function \( g \) first with the input \( 0 \). This gives us a value \( 2 \), which is then used as the input for the function \( f \). From the table, \( f(2) \) evaluates to \( 3 \), so \( f(g(0)) \) yields \( 3 \).

Understanding function evaluation involves keeping track of each function's operations and values, often switching inputs and outputs as needed. This skill is essential in mathematics, particularly in higher-level topics, due to the frequent use of function compositions.

Mathematical Table

A mathematical table is a tool that provides values for different records, organized in rows and columns. In the context of functions, such tables list input-output pairs, which can represent functions graphically or numerically.

For the exercise provided, we use a table that consists of three rows: the first row for \( x \) (inputs), the second row for \( f(x) \) (outputs of function \( f \)), and the third row for \( g(x) \) (outputs of function \( g \)). By reading across the rows, we find the outputs for given inputs.

Tables simplify the process of finding function values without needing to individually perform calculations or apply formulas. They are particularly useful in complex problems involving multiple steps or sets of data.

Step-by-Step Calculation

In mathematics, a step-by-step calculation is a methodical way to work through a problem. It involves tackling each small part of the problem individually before combining them to reach a final solution.

Take the problem of finding \( f(g(2)) \) from the exercise. Here's how you break it down:

• First, locate \( g(2) \) using the corresponding table row. This yields \( 5 \).
• Using the value from \( g(x) \), move to the table for \( f(x) \) and find \( f(5) \), which results in \( 11 \).

Thus, the computation shows that \( f(g(2)) = 11 \).

Step-by-step calculations promote clarity and minimize errors, making complex tasks more manageable. This approach helps students build a strong foundation in problem-solving by understanding every part of the process.