Problem 1 - A Book of Abstract Algebra

Question

\(f: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(f(x)=\sin x\) \(g: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(g(x)=e^{x}\). Find \(f \circ g\) and \(g \circ f\)

Step-by-Step Solution

Answer

\( f \circ g(x) = \sin(e^x) \), and \( g \circ f(x) = e^{\sin x} \).

Step 1: Understand Composition of Functions

The composition of two functions, say \( f \) and \( g \), is denoted by \( f \circ g \), which means that you first apply \( g \) and then apply \( f \) to the result. Therefore, to find \( f \circ g \), you substitute \( g(x) \) into \( f \). Likewise, \( g \circ f \) means applying \( f \) first and then \( g \).

Step 2: Find \(f \circ g\)

We seek to find \( f(g(x)) \). Since \( g(x) = e^x \), substitute \( e^x \) into the function \( f(x) = \sin x \). Thus, \( f(g(x)) = f(e^x) = \sin(e^x) \).

Step 3: Find \(g \circ f\)

Now, find \( g(f(x)) \). With \( f(x) = \sin x \), substitute \( \sin x \) into \( g(x) = e^x \). Thus, \( g(f(x)) = g(\sin x) = e^{\sin x} \).

Step 4: Conclusion

The functions composed are: \( f \circ g = \sin(e^x) \) and \( g \circ f = e^{\sin x} \). These compositions are fully determined by substituting the appropriate expressions into each function.

Key Concepts

Understanding the Sine Function

The sine function, represented as \( \sin(x) \), is a fundamental wave-like function in mathematics. It is periodic, with a consistent pattern repeating every \(2\pi\) units.

• The sine function is defined for all real numbers \(x\).
• The values of \(\sin(x)\) range between \(-1\) and \(1\).
• It is an example of a trigonometric function, often used in relation to triangles and waves.

When graphing \( \sin(x) \), you'll notice smooth curves that go up and down, crossing the x-axis at key points (multiples of \(\pi\)). iIn practical terms, the sine function can represent many real-world periodic processes, such as sound waves or the oscillations of a pendulum.

Exploring the Exponential Function

The exponential function, denoted as \(e^x\), is a mathematical expression where the constant \(e\) (approximately 2.718) is raised to the power of \(x\). This function models continuous growth or decay.

• The base \(e\) is known as Euler's number, a fundamental constant in mathematics.
• \(e^x\) is defined for all real numbers, \(x\).
• The output of the exponential function is always positive, ranging from 0 to infinity.

The exponential function is unique because it grows very quickly and has the incredible property of its proportionality to its own growth rate. This makes it indispensable in fields like finance (for modeling compound interest) and the natural sciences (for describing population growth and radioactive decay).

The Nature of Real Numbers

Real numbers are a set of numbers that include both rational numbers (fractions) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). They cover all points on the number line where each real number can be located at a specific position on this line.

• Real numbers include integers (e.g., \(-3, 0, 5\)), fractions (e.g., \(3/4, -1/2\)), and decimals (e.g., \(2.5, -0.1\)).
• The set of real numbers is denoted by \(\mathbb{R}\).
• Real numbers are vital for counting, measuring, and describing quantities accurately.

Real numbers are critical in calculus and analysis, providing a foundation for studying continuous change. Without real numbers, many practical and theoretical challenges in mathematics would remain unsolved.

Function Operations and Composition

Function operations, particularly composition, involve creating new functions by combining existing ones. The composition of two functions \(f(x)\) and \(g(x)\) is denoted by \(f \circ g(x)\), and it represents the operation of applying \(g(x)\) first and \(f(x)\) thereafter.

• To compose functions, you substitute one function's output into another function.
• \(f \circ g(x)\) means you evaluate \(g(x)\) first, then substitute the result into \(f(x)\).
• This operation creates a new single function from two operations.

For example, given \(f(x) = \sin(x)\) and \(g(x) = e^x\), finding \(f \circ g(x)\) involves evaluating \(g(x)\) which gives \(e^x\), and then substituting \(e^x\) into \(f(x)\) to obtain \(\sin(e^x)\). By understanding function composition, mathematicians can explore more complex relationships and transformations of functions.