Problem 35 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The sum, difference, and product of the functions f and g are \(2x^2 + 1\), \(2x^2 - 2x - 4\), and \(2x^3 + x^2 - 4x -3\) respectively, and their domains are all real numbers. The quotient of the functions is \(\frac{2x^2 - x - 3}{x + 1}\) and its domain is all real numbers except for -1.

Step 1: Find f + g

The sum of the functions f and g, denoted by f+g, is simply their algebraic sum. So, \(f(x) + g(x) = (2x^2 - x - 3) + (x + 1) = 2x^2 + 1\). The domain of f+g is all real numbers because there are no values of x that would make the expression undefined.

Step 2: Find f - g

The difference of the functions f and g, denoted by f-g, is their algebraic difference. So, \(f(x) - g(x) = (2x^2 - x - 3) - (x + 1) = 2x^2 - 2x - 4\). The domain of f-g is all real numbers because there are no values of x that would make the expression undefined.

Step 3: Find fg

The product of the functions f and g, denoted by fg, is their algebraic product. So, \(f(x)g(x) = (2x^2 - x - 3)(x + 1) = 2x^3 + 2x^2 - x^2 - x - 3x - 3 = 2x^3 + x^2 - 4x -3\). The domain of fg is all real numbers because there are no values of x that would make the expression undefined.

Step 4: Find \(\frac{f}{g}\)

The quotient of the functions f and g, denoted by \(\frac{f}{g}\), is \(\frac{f(x)}{g(x)} = \frac{2x^2 - x - 3}{x + 1}\).In general, the domain of a rational function, such as this one, includes all real numbers except for those that make the denominator zero. So, the domain of \(\frac{f}{g}\) is all real numbers except for x = -1, because that would make g(x) = 0 and division by zero is undefined.

Key Concepts

Sum of Functions

Adding two functions together forms a new function that can provide insights into how the original functions relate to each other. The sum of two functions, typically denoted as \( f + g \), is calculated by adding their algebraic expressions together.
In this exercise, given \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), the operation goes as follows:

• Combine like terms: \( (2x^2 - x - 3) + (x + 1) \).
• Simplify: \( 2x^2 - x + x - 3 + 1 = 2x^2 + 1 \).

The result, \( f + g = 2x^2 + 1 \), is defined for all real numbers because no operations in this expression can lead to undefined values.
The domain for both original and resulting functions is simply all real numbers, \( (-\infty, \infty) \).
Understanding this concept allows us to seamlessly handle function additions in algebra.

Difference of Functions

The difference between two functions results in a new function that represents the algebraic subtraction of one from the other. Represented as \( f - g \), this operation helps to see how two functions negate or differ from each other.
For the given functions \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), here's how subtraction is performed:

• Subtract \( g(x) \) from \( f(x) \): \( (2x^2 - x - 3) - (x + 1) \).
• Simplify: \( 2x^2 - x - x - 3 - 1 = 2x^2 - 2x - 4 \).

The resulting function \( f - g = 2x^2 - 2x - 4 \) is also defined for all real numbers, as it contains no operations that could yield undefined outcomes. The domain remains \( (-\infty, \infty) \).
This concept is crucial for problems where function values need to be compared or evaluated against one another.

Product of Functions

The product of two functions creates a representation of how they interact multiplicatively. When you multiply two functions, noted as \( fg \), it demonstrates how their outputs combine to form a new output.
For \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), multiplication is performed as:

• Multiply the functions: \( (2x^2 - x - 3)(x + 1) \).
• Expand the expression: \( 2x^3 + 2x^2 - x^2 - x - 3x - 3 \).
• Simplify: \( 2x^3 + x^2 - 4x - 3 \).

The product \( fg = 2x^3 + x^2 - 4x - 3 \) is valid across all real numbers as there is no potential for division by zero or square roots of negative numbers in this expression. Hence, the domain is \( (-\infty, \infty) \).
This serves as an important exercise to understand the nature of function interactions in a multiplicative context.

Quotient of Functions

The quotient of two functions explores the division aspect, noted as \( \frac{f}{g} \). It reflects how one function can proportionally relate to another by division.
Let's break down the exercise for \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \):

• Formulate the division: \( \frac{f(x)}{g(x)} = \frac{2x^2 - x - 3}{x + 1} \).
• Identify potential issues: The denominator \( x + 1 \) is zero when \( x = -1 \).

The domain of the quotient, therefore, excludes \( x = -1 \) to prevent undefined behavior (division by zero). The valid domain is \( (-\infty, -1) \cup (-1, \infty) \).
Grasping this concept is essential when dealing with rational expressions and understanding where they cease to exist or take on undefined values.