Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Two functions \(f\) and \(g\) can be combined by the arithmetic operations of ______ , ______, _____ , and _______ to create new functions.

Step-by-Step Solution

Answer

The two functions \(f\) and \(g\) can be combined by the arithmetic operations of addition, subtraction, multiplication, and division to create new functions

Step 1: Identify the basic arithmetic operations

The basic arithmetic operations that can be used are addition, subtraction, multiplication, and division.

Step 2: Define the combinations

The combinations of the functions \(f\) and \(g\) using these operations are as follows:1) Addition: \(f + g\), represented as \((f+g)(x)=f(x)+g(x)\), where \(x\) is any real number.2) Subtraction: \(f - g\), represented as \((f-g)(x)=f(x)-g(x)\).3) Multiplication: \(f \cdot g\), represented as \((f \cdot g)(x) = f(x) \cdot g(x)\).4) Division: \(f/g\), represented as \((f/g)(x) = f(x)/g(x)\) (as long as \(g(x) \neq 0)\).

Key Concepts

Addition of Functions

When you add two functions together, you're combining their outputs to form a new function. Let's say we have two functions, \( f(x) \) and \( g(x) \). The addition of these functions is written as \((f + g)(x)\), which simply means you add the outputs of each function for any input \( x \).

So, \((f + g)(x) = f(x) + g(x)\). This operation is straightforward: whatever \( f(x) \) and \( g(x) \) equal individually at any \( x \), you just add those results together.

An example could be \( f(x) = 2x \) and \( g(x) = 3 \). Then \((f + g)(x) = 2x + 3 \). This is a great way to combine functions to create a new function rule that incorporates the behavior of both original functions.

Subtraction of Functions

Subtracting functions is quite similar to addition, but instead of combining outputs, you are calculating their difference. If you have functions \( f(x) \) and \( g(x) \), the subtraction can be expressed as \((f - g)(x) = f(x) - g(x)\).

This means for any input \( x \), subtract the output of \( g \) from the output of \( f \). Again, let's take example functions: \( f(x) = 4x^2 \) and \( g(x) = x \). If you perform the subtraction, \((f-g)(x) = 4x^2 - x\).

It's important to remember that subtraction is not commutative; \( f(x) - g(x) \) will yield a different result than \( g(x) - f(x) \). So pay careful attention to the order of subtraction. This operation helps to analyze the differences between two functional behaviors.

Multiplication of Functions

When multiplying two functions, you're essentially calculating the product of their outputs. So for functions \( f(x) \) and \( g(x) \), the multiplication is shown as \((f \cdot g)(x) = f(x) \cdot g(x)\).

This means for any input \( x \), you take the value of \( f(x) \) and multiply it by \( g(x) \). For example, let \( f(x) = x+1 \) and \( g(x) = x^2 \). Then the product \((f \cdot g)(x) = (x+1) \cdot x^2\).

Multiplying functions can be useful when you want to scale one function by the value of another, or when combining behaviors that are dependent on each other. It's a handy tool to synthesize more complex functions from simpler ones.

Division of Functions

Division of functions involves forming a new function by dividing the output of one function by another, provided the divisor is not zero. If you have \( f(x) \) and \( g(x) \), the division is defined as \((f/g)(x) = \frac{f(x)}{g(x)}\) as long as \( g(x) eq 0 \).

This restriction is crucial; division by zero is undefined in mathematics, so you must ensure \( g(x) \) does not equal zero in your domain of interest. If \( f(x) = x^3 \) and \( g(x) = x-2 \), then \((f/g)(x) = \frac{x^3}{x-2}\), and this is only valid where \( x eq 2 \).

Dividing functions is particularly useful for creating rates or comparing changes between two quantities programmed by the functions. This operation helps students grasp concepts of growth rates and hierarchical relationships between functionalities.