Problem 68 - Calculus

Question

Complete the following table with values for the functions \(f, g,\) and \(h,\) given that: (a) \(f\) is an even function. (b) \(g\) is an odd function. (c) \(h\) is the composition \(h(x)=g(f(x))\) $$\begin{array}{r|c|c|c}\hline x & f(x) & g(x) & h(x) \\\\\hline-3 & 0 & 0 & \\\\-2 & 2 & 2 & \\\\-1 & 2 & 2 & \\\0 & 0 & 0 & \\\1 & & & \\\2 & & & \\\3 & & & \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Fill in missing values using properties of even, odd functions to complete the table.

Step 1: Understanding Even and Odd Functions

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. A function is odd if it satisfies the condition \( g(-x) = -g(x) \). We will use these properties to fill in the missing values for \( f(x) \) and \( g(x) \).

Step 2: Filling Values for the Even Function \( f \)

Since \( f \) is an even function, and we know \( f(-2) = 2 \) and \( f(-1) = 2 \), it follows that \( f(2) = f(-2) = 2 \) and \( f(1) = f(-1) = 2 \). Similarly, since \( f(-3) = 0 \), we have \( f(3) = 0 \).

Step 3: Filling Values for the Odd Function \( g \)

Since \( g \) is an odd function, we use \( g(-2) = 2 \) to find \( g(2) = -g(-2) = -2 \). For \( g(-3) = 0 \), by the properties of an odd function, \( g(3) = -g(-3) = 0 \). Similarly, from \( g(-1) = 2 \) we find \( g(1) = -g(-1) = -2 \).

Step 4: Calculating \( h(x) = g(f(x)) \)

We use the values of \( f(x) \) as inputs to the function \( g \to g(f(x)) \). For example:- For \( x = 1 \), \( h(1) = g(f(1)) = g(2) = -2 \).- For \( x = 2 \), \( h(2) = g(f(2)) = g(2) = -2 \).- For \( x = 3 \), \( h(3) = g(f(3)) = g(0) = 0 \).

Step 5: Completed Table

With the calculations done, here is the completed table:$$\begin{array}{r|c|c|c}\hline x & f(x) & g(x) & h(x) \hline -3 & 0 & 0 & 0 \ -2 & 2 & 2 & 2 \ -1 & 2 & 2 & 2 \ 0 & 0 & 0 & 0 \ 1 & 2 & -2 & -2 \ 2 & 2 & -2 & -2 \ 3 & 0 & 0 & 0 \ \hline \end{array}$$

Key Concepts

Function Composition

Function composition is a process of combining two or more functions to create a new function. In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), the composition of these functions is written as \( g(f(x)) \). This notation means you first apply \( f \) to \( x \), and then apply \( g \) to the result of \( f(x) \).

In the given exercise, the function \( h(x) \) is defined as \( h(x) = g(f(x)) \). This means the value of \( h(x) \) depends not only on \( x \) but on the values of both \( f \) and \( g \).

To calculate \( h(x) \), follow these steps:

• Find \( f(x) \) for the given \( x \).
• Use the result of \( f(x) \) as the input to \( g \).
• The output of \( g(f(x)) \) is your \( h(x) \).

Understanding the composition of functions is essential for solving problems involving multiple function layers.

Function Table

A function table is a structured way to display the relationship between inputs and outputs of various functions. It's especially useful when dealing with multiple functions, as it organizes data in a way that's easy to analyze and understand.

In the given exercise, the function table shows values for \( f(x) \), \( g(x) \), and \( h(x) \) for different values of \( x \). This helps in visualizing how each function interacts with the others.

• The first column lists the input values for \( x \).
• The subsequent columns provide the corresponding values of \( f(x) \), \( g(x) \), and \( h(x) \).

Using a function table can make it much easier to see patterns and relationships, which might not be as obvious when only dealing with function equations. It allows for a clear, side-by-side comparison and can simplify the process of determining how changing one variable affects others.

Even Function Properties

An even function is characterized by symmetry around the y-axis. Mathematically, this means \( f(x) = f(-x) \) for all \( x \) in the domain of the function.

This symmetry implies that for any input \( x \) and its opposite \(-x\), the output of the function is the same. Therefore, once you know the value of \( f(x) \) for a positive \( x \), you automatically know the value for the corresponding negative \( x \).

Key attributes of even functions include:

• Graphical symmetry about the y-axis. If you fold the graph along the y-axis, the two halves will match.
• Examples of even functions include \( f(x) = x^2 \) or \( f(x) = \cos(x) \).

Recognizing the properties of even functions can make solving exercises like the one given much easier, as it allows for reduced calculations by leveraging symmetry.