Problem 10 - Thomas Calculus

Question

Let \(f(x)=x-3, g(x)=\sqrt{x}, \quad h(x)=x^{3},\) and \(j(x)=2 x .\) Express each of the functions in Exercises 9 and 10 as a composite involving one or more of \(f, g, h,\) and \(j .\) $$ \begin{array}{ll}{\text { a. } y=2 x-3} & {\text { b. } y=x^{3 / 2}} \\\ {\text { c. } y=x^{9}} & {\text { d. } y=x-6} \\ {\text { e. } y=2 \sqrt{x-3}} & {\text { f. } y=\sqrt{x^{3}-3}}\end{array} $$

Step-by-Step Solution

Answer

a. \( f(j(x)) \), b. \( g(h(x)) \), c. \( h(h(x)) \), d. \( f(f(x)) \), e. \( j(g(f(x))) \), f. \( g(f(h(x))) \).

Step 1: Express y = 2x - 3

To express the function \( y = 2x - 3 \) in terms of the given functions, we start by observing the components: multiplication and subtraction. We recognize that \( j(x) = 2x \) represents the multiplication by 2, and \( f(x) = x - 3 \) represents subtraction by 3. The composite function representing \( y = 2x - 3 \) can be written as \( f(j(x)) \), where \( j(x) = 2x \).

Step 2: Express y = x^{3/2}

The function \( y = x^{3/2} \) can be expressed by recognizing layers of composition: first taking the cube and then the square root. Therefore, we use \( h(x) = x^3 \) and \( g(x) = \sqrt{x} \). The composite function \( g(h(x)) \) represents this, as \( h(x) = x^3 \) is applied first, followed by \( g(x) = \sqrt{x} \).

Step 3: Express y = x^9

The function \( y = x^9 \) is a result of cubing \( x^3 \). Hence, we use \( h(x) = x^3 \) and create the composite function \( h(h(x)) \), which first cubes \( x \), and then cubes the result again, giving \( x^9 \).

Step 4: Express y = x - 6

The function \( y = x - 6 \) can be broken down into subtracting 3 twice using \( f(x) = x - 3 \). Thus, the composite function can be represented as \( f(f(x)) \), where \( x - 3 \) is applied initially, followed by another subtraction of 3 with the same function.

Step 5: Express y = 2\sqrt{x-3}

The function \( y = 2\sqrt{x-3} \) involves taking a square root and then scaling it by a factor of 2. First, we "shift" \( x \) using \( f(x) = x - 3 \), then we apply \( g(x) = \sqrt{x} \) to get the square root, and finally accelerate the outcome with \( j(x) = 2x \). This can be represented as a composite \( j(g(f(x))) \).

Step 6: Express y = \sqrt{x^3 - 3}

The function \( y = \sqrt{x^3 - 3} \) first involves cubing the value of \( x \) using \( h(x) = x^3 \), subtracting 3 with \( f(x) = x - 3 \), and finally taking the square root with \( g(x) = \sqrt{x} \). The composite form is then \( g(f(h(x))) \), where \( h(x) \) is performed first, followed by \( f(x) \), and then \( g(x) \).

Key Concepts

Function Composition

Function composition is a crucial concept in mathematics, especially when dealing with multiple functions. It allows us to build new functions by applying one function to the results of another. Imagine you have two functions, say \( f(x) \) and \( g(x) \). The composition \( f(g(x)) \) means you apply \( g(x) \) first and then apply \( f(x) \) to the result. This is known as a "nested" function.

In our exercise, we see function compositions like \( f(j(x)) = (x - 3)(2x) \), which first computes \( j(x) = 2x \) and then applies \( f(x) = x - 3 \.\) This sequence of function applications defines a path from the input \( x \) through intermediate values to the final result.

Function compositions are powerful:

• They simplify the description of complex operations using simpler, known functions.
• They express sophisticated processes cleanly, allowing for better understanding and analysis.
• They help decompose problems into manageable segments, making solving them easier.

This methodology is pivotal in many areas of mathematics and applied sciences.

Mathematical Functions

Mathematical functions serve as the backbone of algebra and calculus. A function represents a relationship between a set of inputs and a set of potential outputs. Each input is related to exactly one output. Functions can be expressed as equations like \( f(x) = x - 3 \) or visually represented through graphs.

Functions in the original exercise each serve a distinct purpose:
\( f(x) = x - 3 \) is for subtracting 3 from any input \( x \).
\( g(x) = \sqrt{x} \) calculates the square root of \( x \).
\( h(x) = x^3 \) raises \( x \) to the power of 3, and \( j(x) = 2x \) doubles \( x \).

Collectively, these functions illustrate various operations like subtraction, root extraction, multiplication, and exponentiation. Understanding each function as a "machine" fulfilling a fixed operation helps students recognize patterns and compositions easily without hesitation.

Combining functions (as seen in compositions) unveils intricate relationships and demonstrates how intricate transformations of input data put functions at the heart of the computational and analytical work.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations like addition and multiplication. They are fundamental in representing mathematical situations and enable us to solve problems algebraically. For instance, \( y = 2x - 3 \) is an algebraic expression where \( y \) is defined by subtracting three after multiplying \( x \) by two.

In a composite function, algebraic expressions become especially useful. For example, when expressing \( y = x^{3/2} \), we break down the expression into operations carried out by the functions \( h(x) = x^3 \) and \( g(x) = \sqrt{x} \.\) This expression can be decomposed as \( g(h(x)) \,\) highlighting both cubing the number and then taking its square root.

Algebraic expressions:

• Facilitate the representation of complex mathematical phenomena in a simplified form.
• Help deconstruct operations into manageable parts using known functions.
• Serve as the foundation for equations that can be manipulated to solve problems.

Understanding how to work with algebraic expressions is an empowering skill that lays the groundwork for solving equations and performing various algebraic manipulations.