Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Express each of the following rules for obtaining the output of a function using functional notation. (a) Square the input, add 3 , and take the square root of the result. (b) Double the input, then add 7 . (c) Take half of 3 less than the input. (d) Increase the input by 10, then cube the result.

Step-by-Step Solution

Answer

The functional notations for the given rules are (a) \(f(x) = \sqrt{x^2 + 3}\), (b) \(f(x) = 2x + 7\), (c) \(f(x) = \frac{1}{2}(x-3)\), (d) \(f(x) = (x + 10)^3\)

Step 1: Solution for (a): Square the input, add 3 , and take the square root of the result.

Taking the input as \(x\), the described operations can be denoted as: \(f(x) = \sqrt{x^2 + 3}\)

Step 2: Solution for (b): Double the input, then add 7.

For an input \(x\), the operations to achieve the output can be denoted in functional notation as: \(f(x) = 2x + 7\)

Step 3: Solution for (c): Take half of 3 less than the input.

When \(x\) represents the input, the operations leading to the output based on the given rule are: \(f(x) = \frac{1}{2}(x-3)\)

Step 4: Solution for (d): Increase the input by 10, then cube the result.

Given that \(x\) is the input, this rule's operations can be written in functional notation as: \(f(x) = (x + 10)^3\)

Key Concepts

Defining Functions

The concept of functions is fundamental in mathematics, particularly in algebra. At its core, a function can be thought of as a rule that assigns each input exactly one output. It is often represented by the symbol 'f(x)', where 'f' is the function, and 'x' is the input variable. For instance, in the exercise's solution for (a), the rule given describes a two-step process: squaring the input (denoted as 'x') and then adding three, followed by taking the square root of the sum. This rule is concisely expressed in functional notation as \(f(x) = \sqrt{x^2 + 3}\).

When defining functions, it's important to specify what types of numbers can be used as inputs, known as the domain, and the corresponding range, which comprises all possible outputs. Effective communication in mathematics relies on this unambiguous and standard notation to share ideas universally. In a classroom setting, this clarity is key for students to understand the transformations that any given input undergoes through the function's rule.

Operations on Functions

Operations on functions involve manipulating these mappings to create new functions. These operations are analogous to those performed on numbers, such as addition, subtraction, multiplication, and division. In the textbook exercise, learners are introduced to different operations on a single function. For example, the solution for (b) presents the operation of first doubling the input and then adding seven, which translates to the function \(f(x) = 2x + 7\).

Understanding these operations is vital for students to grasp more complex function topics, such as function composition and inverse functions. Additionally, when functions are combined, they follow properties similar to real numbers—commutative, associative, distributive—which are foundational concepts that help in simplifying and solving complex functional expressions. For instance, recognizing that for the solution provided in (d), the input was increased by 10 before being cubed, we define it as \(f(x) = (x + 10)^3\), thus describing a combination of operations in a specific order.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operation symbols, and sometimes parentheses that represent a quantity or a relationship. Every part of an expression represents a piece of the mathematical puzzle. In the context of functions, expressions can define the rule applied to the input variable to produce outputs. For example, \(f(x) = \frac{1}{2}(x-3)\), from the step-by-step solution to part (c) of the exercise, simplifies a more complex situation into a manageable mathematical expression.

Learning how to correctly form and manipulate these expressions is crucial for students as it increases their problem-solving skills and mathematical fluency. Expressions can be evaluated, simplified, factored, or expanded, allowing for exploration and manipulation to find desired outcomes or prove certain properties. The ability to work comfortably with mathematical expressions empowers students, giving them the tools to tackle advanced concepts in algebra, calculus, and beyond.