Problem 45 - Calculus: One and Several Variables

Question

Find \(f \circ g\) and \(g \circ f\). $$f(x)=1-x^{2}, g(x)=\sin x$$

Step-by-Step Solution

Answer

The composite functions \(f \circ g\) and \(g \circ f\) are \(1- \sin^{2}x\) and \(\sin (1- x^{2})\), respectively

Step 1: Substitute \(g(x)\) in \(f(x)\)

To find \(f \circ g\), substitute \(g(x)\) for \(x\) in \(f(x)\). So, \(f \circ g = f(g(x))= 1- (g(x))^{2} = 1- (\sin x)^{2}\)

Step 2: Simplify the expression

The expression \(1- (\sin x)^{2}\) can be written as \(1- \sin^{2}x\)

Step 3: Substitute \(f(x)\) in \(g(x)\)

Now, to find \(g \circ f\), we need to substitute \(f(x)\) in place of \(x\) in \(g(x)\). Thus, \(g \circ f = g(f(x)) = \sin (f(x)) = \sin (1- x^{2})\)

Step 4: Simplifying the expression

The expression \(\sin (1-x^{2})\) is a simplified expression and cannot be further simplified

Key Concepts

Trigonometric Functions

In mathematics, trigonometric functions are essential tools for analyzing angles and dimensions. Notably, the sine function, denoted as \( \sin(x) \), is one of the primary trigonometric functions. It relates an angle measure to the ratio of the length of the opposite side to the hypotenuse in a right triangle.
Understanding trigonometric functions is crucial as they appear in various mathematical and engineering applications. These functions are periodic, meaning they repeat their values in regular intervals. For the sine function, the fundamental period is \( 2\pi \). This property makes the sine function invaluable in scenarios that involve waves and oscillatory motion, such as sound waves or pendulum movements.
The exercise involves using the sine function in function compositions, showcasing its versatility in mathematical expressions. When substituting the trigonometric function into other functions, it's essential to maintain the properties of the original trigonometric identities, such as \( \sin^2(x) + \cos^2(x) = 1 \). This identity is used to simplify expressions like \( 1 - (\sin(x))^2 \) to fit easily into broader mathematical frameworks.

Substitution Method

The substitution method is a fundamental mathematical technique used to simplify expressions and solve equations. This method involves replacing variables with other expressions that can simplify the solution process.
In function composition, the substitution method plays an integral role. When finding \( f \circ g \), we substitute \( g(x) \) into \( f(x) \). For our exercise, this means replacing \( x \) in \( f(x) = 1 - x^2 \) with \( \sin(x) \), resulting in the expression \( 1 - (\sin(x))^2 \). This substitution simplifies complex problems by breaking them into smaller, more manageable parts.
The method is also used in reverse when calculating \( g \circ f \), whereby \( f(x) \) is substituted back into \( g(x) \). This calculated approach can be particularly helpful when tackling higher mathematics problems, offering a clear path through intricate calculations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value or relationship. In calculus and algebra, expressions are used to define functions and perform operations like addition, multiplication, and exponentiation.
In the exercise, expressions such as \( 1 - (\sin(x))^2 \) and \( \sin(1-x^2) \) are simplified forms derived through function composition. These expressions result from applying operations and trigonometric connections.
Understanding mathematical expressions involves recognizing the underlying principles, such as

• Utilizing algebraic identities and trigonometric relationships.
• Simplifying expressions by applying rules of operation and identities.

In practical scenarios, managing these expressions efficiently can lead to solving complex problems more quickly. This skill is essential in both academic mathematics and real-world applications, enabling clearer solutions and innovative problem-solving approaches.