Problem 9 - College Algebra

Question

Find \(f+g, f-g, f g,\) 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

(f+g)(x) = 2x^2 - 2, the domain is all real numbers. (f-g)(x) = 2x^2 - 2x - 4, the domain is all real numbers. (f g)(x) = 2x^3 + x^2 - 4x - 3, the domain is all real numbers. \(\frac{f}{g}(x) = \frac{2x^2 - x - 3}{x + 1}\), the domain is all real numbers except -1.

Step 1: Function Addition

The sum of the functions f(x) and g(x) can be found by adding the respective terms of each function together:\[ (f+g)(x) = f(x) + g(x) = (2x^2 - x - 3) + (x + 1) = 2x^2 - x + x - 3 + 1 = 2x^2 - 2 \]So, (f+g)(x) = 2x^2 - 2. The domain for this function is all real numbers, as there are no restrictions on the values for x.

Step 2: Function Subtraction

The difference between the functions f(x) and g(x) can be found by subtracting g(x) from f(x):\[ (f-g)(x) = f(x) - g(x) = (2x^2 - x - 3) - (x + 1) = 2x^2 - x - x - 3 - 1 = 2x^2 - 2x - 4\]So, (f-g)(x) = 2x^2 - 2x - 4. The domain for this function is all real numbers, as there are no restrictions on the values for x.

Step 3: Function Multiplication

The product of the functions f(x) and g(x) can be found by multiplying f(x) by g(x):\[(f g)(x) = f(x) g(x) = (2x^2 - x - 3) (x + 1) = 2x^3 + 2x^2 - x^2 - x - 3x - 3 = 2x^3 + x^2 - 4x - 3\]So, (f g)(x) = 2x^3 + x^2 - 4x - 3. The domain for this function is all real numbers, as there are no restrictions on the values for x.

Step 4: Function Division

The quotient of the functions f(x) and g(x) can be found by dividing f(x) by g(x):\[\frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{2x^2 - x - 3}{x + 1}\]There is a limit to the values for x in this case. x cannot be -1, as it would make the denominator zero, undefined. Therefore, the domain of this function is all real numbers except -1.

Key Concepts

Function Addition

Function Addition involves combining two functions, say \( f(x) \) and \( g(x) \), to form a new function \((f+g)(x)\). This new function is obtained by adding the outputs of \( f(x) \) and \( g(x) \) for the same input \( x \).
For instance, if \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), then:

• Add each term separately: \((f+g)(x) = (2x^2 - x - 3) + (x + 1)\)
• Simplify: \((f+g)(x) = 2x^2 - x + x - 3 + 1 = 2x^2 - 2\)

The resultant function, \((f+g)(x)\), is quadratic. The domain includes all real numbers since no restrictions arise from addition.

Function Subtraction

Function Subtraction involves finding the difference between function outputs. To subtract \( g(x) \) from \( f(x) \), we calculate \((f-g)(x)\). It's as simple as subtracting the function expressions term by term.
Using \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), we have:

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

The domain remains all real numbers, allowing any value of \( x \) without restrictions.

Function Multiplication

Function Multiplication creates a new function by multiplying the outputs of \( f(x) \) and \( g(x) \) for each input \( x \). Multiply each term of one function by every term of the other.
Given \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \), the product is calculated as:

• Multiply each term: \((fg)(x) = (2x^2 - x - 3)(x + 1) = 2x^3 + 2x^2 - x^2 - x - 3x - 3 \)
• Combine like terms: \((fg)(x) = 2x^3 + x^2 - 4x - 3 \)

This function's domain, like the previous functions, includes all real numbers, owing to the absence of limitations.

Function Division

Function Division involves dividing the expression \( f(x) \) by \( g(x) \) to form \( \frac{f}{g}(x) \). But be careful! Ensure the denominator \( g(x) \) isn't zero, as division by zero is undefined.
Consider \( f(x) = 2x^2 - x - 3 \) and \( g(x) = x + 1 \):

• Divide the functions: \( \frac{f}{g}(x) = \frac{2x^2 - x - 3}{x + 1} \)
• Identify restrictions: \( x + 1 eq 0 \), hence \( x eq -1 \)

The domain of this function excludes \(-1\), meaning it's defined for all real numbers except \(-1\).

Domain and Range

Determining the domain and range for functions is crucial to understanding where the function is applicable (domain) and what values it covers (range).

• Domain: The set of all possible inputs \( x \) for which the function is defined. For example, function division requires that the denominator be non-zero. Hence, \( \frac{2x^2 - x - 3}{x + 1} \) has a domain of all real numbers except \( x = -1 \).
• Range: The set of all potential outputs \( f(x) \) or \( g(x) \) from a function for the values in the domain. The range depends on the nature of the function such as quadratic or linear.

Adhering to these concepts ensures proper understanding and handling of functions, especially when performing operations like addition, subtraction, multiplication, and division.