Problem 35 - Precalculus Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{x}\). Determine the domain for each function. $$f(x)=2 x^{2}-x-3, g(x)=x+1$$

Step-by-Step Solution

Answer

The sum of the functions \(f+g\) is \(2x^2 + 1\) with domain being all real numbers. The difference \(f-g\) is \(2x^2 - 2x - 4\) with domain being all real numbers. The product of the functions \(f \cdot g\) is \(2x^3 - x^2 - 3x + 1\) with domain being all real numbers. The quotient \(f/g\) cannot be simplified further and its domain is all real numbers except for -1.

Step 1: Find the Sum of Functions (f+g)

To find \(f+g\), add the two given functions together: \(f(x) + g(x) = (2x^2 - x - 3) + (x + 1)\). Simplifying this gives \(f+g = 2x^2 + 1\). The domain of \(f+g\) is all real numbers since every real number can be plugged into \(f+g\) and return a real number output.

Step 2: Find the Difference of Functions (f-g)

To find \(f-g\), subtract the second function from the first one: \(f(x) - g(x) = (2x^2 - x - 3) - (x + 1)\). Simplifying this gives \(f-g = 2x^2 - 2x - 4\). The domain of \(f-g\) is also all real numbers as every real number can be plugged into \(f-g\) without causing any restrictions.

Step 3: Find the Product of Functions (f * g)

To find \(f * g\), multiply the two functions together: \(f(x) * g(x) = (2x^2 - x - 3) * (x + 1)\). Simplifying this gives \(f * g = 2x^3 - x^2 - 3x + 1\). The domain of \(f * g\) is all real numbers as every real number can be plugged into \(f * g\).

Step 4: Find the Quotient of Functions (f/g)

To find \(f/g\), divide the first function by the second: \(f(x) / g(x) = (2x^2 - x - 3) / (x + 1)\). In this case, the expression cannot be simplified further. The domain of \(f/g\) is all real numbers except for -1, because if x = -1, then the denominator becomes zero, leading to an undefined expression.

Key Concepts

Domain of Functions

When we talk about the domain of a function, we are referring to the set of all possible input values (often represented by "x") that allow the function to produce valid outputs. It’s a key aspect of understanding how functions behave.
For most polynomial functions like those given in this exercise, the domain is all real numbers, unless there's division involved, which could impose restrictions. For example, in the quotient of functions, the domain excludes values that make the denominator zero because dividing by zero is undefined.

• Example: The function \( f/g \) is undefined at \( x = -1 \), thus \( x = -1 \) is not part of its domain.

Knowing the domain is essential for evaluating the function accurately and ensuring the solutions make mathematical sense.

Sum of Functions

The sum of functions involves adding two functions together. Given two functions \( f(x) \) and \( g(x) \), the sum \( f+g \) is calculated by adding their expressions together.
In this exercise, the function \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \) are added to create \( f+g = 2x^2 + 1 \). This results from carefully combining like terms and simplifying the expression.

• The domain of \( f+g \) is all real numbers. There are no restrictions because none of the individual functions has denominators or other limiting behavior.

Understanding how to sum functions helps in forming new functions from existing ones, expanding the ways we can manipulate and understand mathematical equations.

Difference of Functions

Finding the difference of functions involves subtracting one function from another. The process is quite straightforward and similar to the sum of functions but involves a subtraction.For \( f-g \), where \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), you subtract \( g(x) \) from \( f(x) \), which gives \( f-g = 2x^2 - 2x - 4 \) after combining like terms and simplifying the expression.

• This operation also results in a function with a domain of all real numbers, since subtraction here introduces no constraints.

Mastering subtraction of functions provides a foundation for understanding more complex operations and their respective impacts on function behavior.

Product of Functions

The product of functions involves multiplying two functions together. This operation combines the terms of each function through multiplication.
In the provided problem, multiply \( f(x) = 2x^2 - x - 3 \) with \( g(x) = x + 1 \) to obtain the product \( f \cdot g = 2x^3 - x^2 - 3x + 1 \). Each term from one function gets multiplied by each term from the other, and the resulting terms are then combined and simplified.

• The domain of \( f \cdot g \), like those of addition and subtraction, encompasses all real numbers as multiplication of polynomials doesn't create undefined values inherently.

Understanding the product of functions is crucial as it forms the basis for many mathematical and real-world applications where functions describe complex relationships.

Quotient of Functions

The quotient of functions requires division of one function by another, giving us \( \frac{f}{g} \). This operation is vital but requires special attention regarding the domain. Division by zero is not defined, so any value that makes the denominator zero must be excluded from the domain.
For the given functions \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), the quotient is \( \frac{f}{g} = \frac{2x^2 - x - 3}{x + 1} \).

• The domain here is all real numbers except \( x = -1 \), since at this value the denominator becomes zero, rendering the function undefined.

Grasping quotients of functions is essential as it helps identify how relationships change at specific function inputs, especially where potential discontinuities arise.