Problem 61 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=\frac{1}{\left(x^{2}+9\right)^{3 / 2}}\) and \(g(x)=3 \tan x,\) show that\((f \circ g)(x)=\frac{1}{27\left|\sec ^{3} x\right|}\)

Step-by-Step Solution

Answer

\( (f \circ g)(x) = \frac{1}{27 \left| \sec^{3} x \right|} \)

Step 1 - Understand the functions

Given functions: 1. Function f(x) is defined as \( f(x)= \frac{1}{(x^{2}+9)^{3/2}}\)2. Function g(x) is defined as \( g(x)=3 \,\tan x\)

Step 2 - Find the composite function

The composite function \((f \circ g)(x)\) means we use \( g(x)\) as the input for \( f(x)\). Thus, substitute \( g(x) = 3 \, \tan x \) into \( f(x) \), resulting in \( f(g(x)) = f(3 \, \tan x) \)

Step 3 - Substitute and simplify

Substitute 3 \( \tan x \) into \( f(x) \):\[ f(3 \, \tan x) = \frac{1}{((3 \, \tan x)^{2} + 9)^{3/2}} \]Simplify the expression within the denominator:\[ (3 \, \tan x)^{2} + 9 = 9 \, \tan^{2} x + 9 = 9 (\tan^{2} x + 1) \]Use the trigonometric identity \( \tan^{2} x + 1 = \sec^{2} x \):\[ 9 (\tan^{2} x + 1) = 9 \, \sec^{2} x \]Substitute back into the function:\[ f(3 \, \tan x) = \frac{1}{(9 \, \sec^{2} x)^{3/2}} \]

Step 4 - Simplify further

Simplify the exponent in the denominator:\[ (9 \, \sec^{2} x)^{3/2} = 9^{3/2} \, \sec^{3} x = (3^{2})^{3/2} \, \sec^{3} x = 3^{3} \, \sec^{3} x = 27 \, \sec^{3} x \]So, we have:\[ f(3 \, \tan x) = \frac{1}{27 \, \sec^{3} x} \]

Step 5 - Include absolute value

Since we must account for the absolute value in the solution:\[ f(3 \, \tan x) = \frac{1}{27 \, | \sec^{3} x | } \]Thus, \(\ (f \circ g)(x) = \frac{1}{27 | \sec^{3} x | } \)

Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the result of another function. If you have two functions, say \(f(x)\) and \(g(x)\), the composite function \((f \, \circ \, g)(x)\) means you're plugging \(g(x)\) into \(f(x)\). For example, if \(f(x) = \frac{1}{(x^{2}+9)^{3/2}}\) and \(g(x) = 3 \tan x\), the composite function is found by substituting \(3 \tan x\) into \(f(x)\). So, the composite function is:\[ (f \, \circ \, g)(x) = f(g(x)) = f(3 \tan x) \]Composition of functions allows us to combine two functions into one, making it easier to work with complex mathematical expressions.

Trigonometric Identities

Trigonometric identities are equations that hold for all values of the variable where both sides of the equation are defined. These identities help simplify trigonometric expressions and solve trigonometric equations. One commonly used identity is:\[ \tan^2 x + 1 = \sec^2 x \]In our problem, we use this identity to simplify the expression inside the composite function. When substituting \( g(x) = 3 \tan x \) into \( f(x) \), we arrive at:\[ (3 \tan x)^2 + 9 = 9(\tan^2 x + 1) = 9 \cdot \sec^2 x \]Utilizing trigonometric identities can significantly reduce the complexity of expressions, making them easier to solve.

Simplification of Expressions

Simplification of expressions involves reducing them to their most basic form. This often includes combining like terms, using identities, and performing arithmetic operations. For our function, after substituting and using trigonometric identities, we simplify the denominator:\[ 9(\sec^2 x)^{3/2} = (9 \sec^2 x)^{3/2} = 9^{3/2} \cdot \sec^3 x \]Breaking it down further:\[ 9^{3/2} = (3^2)^{3/2} = 3^3 = 27 \]Thus:\[ (9 \sec^2 x)^{3/2} = 27 \sec^3 x \]Simplifying mathematical expressions is a crucial step in solving complex problems. It helps to see patterns and make connections between different parts of the problem.

Absolute Value in Functions

The absolute value of a number is its distance from zero on the number line, regardless of direction. In functions, the absolute value ensures that the output is always non-negative. For example:\( | -5 | = 5 \) and \( | 5 | = 5 \).In the context of our function, the absolute value appears as \( | \sec^3 x | \). This is because \( \sec x \) can be positive or negative depending on the value of \( x \). The use of absolute value ensures that any negative value is turned positive:\[ f(3 \tan x) = \frac{1}{27 | \sec^3 x |} \]This is important to get the correct result since mathematical rules need to account for all possible values of \( x \). Incorporating absolute value helps maintain the integrity and correctness of the solution.