Problem 9 - Calculus: One and Several Variables

Question

Determine \(f+g, f-g , f \cdot g , f / g,\) and give the domain of each. $$f(x)=2 x-3, g(x)=2-x$$

Step-by-Step Solution

Answer

The sum of the functions is \(f+g = x+5\) with a domain of all real numbers. The difference of the functions is \(f-g = x+1\) with a domain of all real numbers. The product of the functions is \(f \cdot g = -2x^{2} + 7x - 6\) with a domain of all real numbers. Finally, the quotient of the functions is \(f/g = (2x-3) / (2-x)\) with a domain of all real numbers except \(2\).

Step 1: Determining \(f+g\)

First, find the sum of functions f and g:\(f+g=f(x)+g(x)\)=(2x-3)+(2-x). After simplifying, the sum of the functions will be \(f+g=2x-x+2+3=x+5.\)

Step 2: Determining \(f-g\)

Next, find the difference of f and g: \(f-g=f(x)-g(x)=(2x-3)-(2-x). After simplifying, the difference of the functions will be \(f-g=2x-(-x)+3-2=x+1\).

Step 3: Determining \(f \cdot g\)

Now, find the product of f and g: \(f \cdot g = f(x) \cdot g(x) = (2x-3) \cdot (2-x). After multiplying, \(f \cdot g = 2x(2)+2x(-x)-3 \cdot 2-3 (-x) = 4x - 2x^{2} - 6 + 3x = -2x^{2} + 7x - 6.\)

Step 4: Determining \(f/g\) and its domain

Finally, find the division of f by g: \(f/g = f(x) / g(x) = (2x-3) / (2-x). In this part, we have to be careful with the domain. The function \(f/g\) is not defined when the denominator equals zero. Solve for \(x\) in the equation \(g(x)=2-x=0\) gives us \(x=2\). So, the domain of \(f/g\) is all real numbers except \(2\).

Key Concepts

Function Operations

When working with functions in algebra, we're not limited to just evaluating them separately. We can perform various operations on functions, much like we do with numbers.

• Addition: To find \(f+g\), we simply add the outputs of functions \ f(x) \ and \ g(x) \ at each point \ x \. For our functions \ f(x) = 2x - 3 \ and \ g(x) = 2 - x \, the addition will be \(f+g = (2x-3) + (2-x) = x+5\).
• Subtraction: Similarly, \(f-g\) involves subtracting the outputs: \(f-g = (2x-3) - (2-x) = x+1\).
• Multiplication: When multiplying functions, \(f \cdot \ g\) is found by multiplying the outputs: \(f \cdot \ g = (2x-3)(2-x) = -2x^2 + 7x - 6\).
• Division: Finally, \(f/g\) is determined by dividing the outputs, \(f/g = \frac{2x-3}{2-x}\), with careful consideration of when \ g(x) \ is not zero.

By mastering these operations, we extend the ways in which we analyze relationships between variables.

Domain of a Function

The domain of a function is essentially the collection of all possible input values (x-values) for which the function will return a valid output. Understanding the domain is crucial because it tells us where a function 'lives' or operates.In our case:

• For \(f+g\), \(f-g\), and \(f \cdot \ g\), there are no special conditions given by the original functions \(f(x) = 2x-3\) and \(g(x) = 2-x\), so their domains are all real numbers.
• However, for the division operation \(f/g\), we need to be careful. The functions in the denominator cannot be zero, \(g(x) = 2-x=0\) which occurs when \(x=2\). Therefore, the domain of \(f/g\) is all real numbers except \(x=2\).

Identifying the domain helps in graphing functions and ensuring calculations remain valid, particularly for operations that can result in undefined values.

Rational Functions

Rational functions, a crucial concept in algebra, are functions formed by a ratio of two polynomials—as in our division operation \(f/g\) example.Characteristics of rational functions include:

• They often exhibit asymptotic behavior, where the graph approaches a line (vertical or horizontal) but never actually meets it. This is typically due to the denominator equating to zero.
• In \(f/g = \frac{2x-3}{2-x}\), the denominator \(2-x\) should not be zero, as it results in an undefined value. This is why \(x=2\) is excluded from the domain.
• Rational functions can have discontinuities (breaks in the graph), often requiring careful analysis to understand their behavior. These occur where a function is not defined.

Understanding rational functions helps us navigate more complex real-world problems, where phenomena can be modeled by one quantity being divided by another, such as rates and proportions.