Problem 3 - Algebra and Trigonometry

Question

If the rule of the function \(f\) is "add one" and the rule of the function \(g\) is "multiply by 2 " then the rule of \(f \circ g\) is " ___________________" and the rule of \(g \circ f\) is " _______________________"

Step-by-Step Solution

Answer

The rule of \(f \circ g\) is 'multiply by 2, then add 1', and for \(g \circ f\) it is 'add 1, then multiply by 2'.

Step 1: Understand Function Composition

When dealing with function composition, remember that \(f \circ g\) represents the function \(f(g(x))\). This means you first apply function \(g\) to \(x\), and then apply function \(f\) to the result.

Step 2: Define the Functions

Let's clearly define each function rule given in the problem. - Function \(f(x)\) is defined by the rule 'add one', so \(f(x) = x + 1\).- Function \(g(x)\) is defined by the rule 'multiply by 2', so \(g(x) = 2x\).

Step 3: Apply Function Composition for \(f \circ g\)

Compute \(f \circ g(x)\) by first applying \(g(x)\) and then \(f(x)\).- First apply \(g(x): g(x) = 2x\).- Now apply \(f\) to the result of \(g(x): f(g(x)) = f(2x) = 2x + 1\).Thus, the rule for \(f \circ g\) is 'multiply by 2, then add 1'.

Step 4: Apply Function Composition for \(g \circ f\)

Compute \(g \circ f(x)\) by first applying \(f(x)\) and then \(g(x)\).- First apply \(f(x): f(x) = x + 1\).- Now apply \(g\) to the result of \(f(x): g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2\).Thus, the rule for \(g \circ f\) is 'add 1, then multiply by 2'.

Key Concepts

Function Operations

Function operations involve different ways of combining or manipulating functions to create new functions. When speaking about operations on functions, you are often thinking about things like addition, subtraction, multiplication, and even division of functions.

Function composition is another significant operation where you take the output from one function and feed it into another. This new function is often denoted by \(f \circ g\), which is read as "f composed with g". It involves finding the function output of \(g(x)\) first and then applying \(f\) to that output. This process can dramatically change the behavior of the resulting function, offering a wide variety of applications in different fields like physics or engineering.

Remember:

• Always apply the function inside the parentheses first.
• The order of function operations matters greatly, as shown by \(f \circ g\) being different from \(g \circ f\).

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of function composition, algebra helps us to combine and simplify these functions using rules and formulas.

When you define a function, such as \ f(x) = x + 1 \, and another function \ g(x) = 2x \, you're leveraging algebraic concepts. These expressions describe actions you perform on an input \(x\).

Using algebraic techniques, such as substitution, you can find out what happens when you combine these functions. For example, inputting \(x\) into \ g(x) \ first leads to \(2x\), and plugging this result into \ f(x) \ results in \(2x + 1\) for \(f \circ g\). Algebra simplifies the process of seeing how inputs move through a series of operations in a composed function.

• Algebra helps in visualizing and simplifying the expressions resulting from function composition.
• It's crucial to practice your algebra skills to easily manipulate and work through function rules.

Mathematical Functions

Mathematical functions are essential tools in mathematics. They enable us to map every input value from a specific set to an output value. These mappings are based on specific rules which define the behavior of the function.

Think of functions as little machines. You input a number, the function processes it according to its rule, and outputs another number. For example, with the function \ f(x) = x + 1 \, you put in \(2\) and get out \(3\).

The essence of function composition relies on these rules interacting with one another. By composing \(f\) and \(g\), you create a new composite function that acts according to a combined set of operations. This process provides insight into more complex relationships between inputs and their transformed outputs.

• Understand each function's rule individually before attempting to compose them.
• Basic functions can be combined to create more advanced hybrid functions, allowing for more complex calculations and solutions.