Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Let \(f(x)=\frac{3 x+2}{x^{2}+1}\) and \(g(x)=\frac{1}{100} \cos (100 x)\). (a) Use functional composition to form \(h(x)=(f \circ g)(x)\), as well as \(j(x)=(g \circ f)(x)\). $$ \begin{array}{l} \text { (a) Use functional composition to form } h(x)=(f \circ g)(x), \text { as }\\\ \text { well as } j(x)=(g \circ f)(x) \text { . } \end{array} $$

Step-by-Step Solution

Answer

\(h(x) = \frac{\frac{3}{100}\cos(100x) + 2}{\frac{1}{10000}\cos^2(100x) + 1}\), \(j(x) = \frac{1}{100} \cos\left(100\cdot \frac{3x + 2}{x^2 + 1}\right)\).

Step 1: Understand the Functional Composition

Functional composition involves applying one function to the results of another function. For \(h(x) = (f \circ g)(x)\), we apply \(g(x)\) first, then \(f\) to the result. For \(j(x) = (g \circ f)(x)\), we apply \(f(x)\) first, then \(g\) to the result.

Step 2: Substitute for h(x)

To find \(h(x) = (f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). This means replacing \(x\) in \(f(x)\) with \(g(x) = \frac{1}{100} \cos (100x)\). So, \(h(x) = f\left(\frac{1}{100}\cos(100x)\right) = \frac{3\left(\frac{1}{100}\cos(100x)\right)+2}{\left(\frac{1}{100}\cos(100x)\right)^2 + 1}\).

Step 3: Simplify h(x)

Calculate the expression for \(h(x)\). The numerator becomes \(\frac{3}{100}\cos(100x) + 2\) while the denominator becomes \(\left(\frac{1}{100}\cos(100x)\right)^2 + 1 = \frac{1}{10000}\cos^2(100x) + 1\). Thus, \(h(x) = \frac{\frac{3}{100}\cos(100x) + 2}{\frac{1}{10000}\cos^2(100x) + 1}\).

Step 4: Substitute for j(x)

To find \(j(x) = (g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). This requires replacing \(x\) in \(g(x)\) with \(f(x) = \frac{3x + 2}{x^2 + 1}\). Thus, \(j(x) = g\left(\frac{3x + 2}{x^2 + 1}\right) = \frac{1}{100} \cos\left(100\cdot \frac{3x + 2}{x^2 + 1}\right)\).

Step 5: Final Expressions

Finally, \(h(x) = \frac{\frac{3}{100}\cos(100x) + 2}{\frac{1}{10000}\cos^2(100x) + 1}\) and \(j(x) = \frac{1}{100} \cos\left(100\cdot \frac{3x + 2}{x^2 + 1}\right)\). These expressions represent the compositions of functions \(h\) and \(j\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It provides tools for understanding and working with functions and their rates of change. In functional composition, calculus can help in analyzing how the composition of two functions behaves, especially how derivatives and integrals can be used to explore their properties.

For example, when dealing with composite functions like \( h(x) = (f \circ g)(x) \) or \( j(x) = (g \circ f)(x) \), calculus allows us to examine:

• The derivative of the composite function, which helps in understanding how the function changes at any point.
• How the behavior of these functions affects graph properties like slopes, tangents, and curvature.

Understanding these concepts is crucial, especially if the functions involved depict real-world situations where change and motion occur, like physics problems or economic models.

Functions

Functions are mathematical entities that describe how one quantity changes with another. In simpler terms, a function takes an input, performs certain operations, and delivers an output. Functional composition involves two functions, say \( f(x) \) and \( g(x) \), where the result of one function becomes the input of another.

This process can be done in different orders. For instance:

• In \( h(x) = (f \circ g)(x) \), you first apply \( g(x) \) and then apply \( f(x) \) on the result of \( g(x) \).
• Conversely, in \( j(x) = (g \circ f)(x) \), you apply \( f(x) \) first and then \( g(x) \) on the result of \( f(x) \).

Each of these paths can yield different results and interpretations, making understanding functions and their compositions a cornerstone in mathematics.

Trigonometric Functions

Trigonometric functions are a subset of functions that relate the angles of a triangle to the lengths of its sides. Common examples include sine, cosine, and tangent functions. In the given problem, the function \( g(x) = \frac{1}{100} \cos(100x) \) incorporates the cosine function.

These functions are pivotal in calculations involving periodic phenomena, such as sound waves, light waves, and circular motion. They exhibit cyclic behavior, which makes them ideal for modeling these types of repeated patterns.

When used in functional compositions:

• For \( h(x) = f(g(x)) \), we see how a trigonometric function is transformed through another function \( f(x) \).
• In \( j(x) = g(f(x)) \), the trigonometric function \( g(x) \) applies to the outcome of the function \( f(x) \), potentially altering the cycle or amplitude of the wave it represents.

Understanding trigonometric functions and their composition can deepen insights into many engineering and physics problems.