Problem 5 - Precalculus Mathematics for Calculus

Question

Find \(f+g, f-g, f g,\) and \(f / g\) and their domains. $$f(x)=x-3, \quad g(x)=x^{2}$$

Step-by-Step Solution

Answer

All functions have domains of all real numbers, except \(f/g\), which excludes \(x=0\).

Step 1: Finding the Sum (\(f+g\))

To find the sum \((f+g)(x)\), we add the functions \(f(x)\) and \(g(x)\): \[(f+g)(x) = f(x) + g(x) = (x - 3) + x^2 = x^2 + x - 3\].This sum function is a polynomial and its domain is all real numbers, \( \mathbb{R} \).

Step 2: Finding the Difference (\(f-g\))

To find the difference \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\):\[(f-g)(x) = f(x) - g(x) = (x - 3) - x^2 = -x^2 + x - 3\]. This difference function is also a polynomial, so its domain is all real numbers, \( \mathbb{R} \).

Step 3: Finding the Product (\(fg\))

To find the product \((fg)(x)\), we multiply the functions \(f(x)\) and \(g(x)\):\[(fg)(x) = f(x) \, g(x) = (x - 3) \, x^2 = x^3 - 3x^2\].This product function is a polynomial, and its domain is all real numbers, \( \mathbb{R} \).

Step 4: Finding the Quotient (\(f/g\))

To find the quotient \((f/g)(x)\), we divide \(f(x)\) by \(g(x)\):\[(f/g)(x) = \frac{f(x)}{g(x)} = \frac{x - 3}{x^2}\].The domain of this quotient function excludes values where \(g(x) = 0\). Since \(g(x) = x^2\), the excluded value is \(x = 0\). Therefore, the domain is all real numbers except \(0\), denoted as \( \mathbb{R} \setminus \{0\} \).

Key Concepts

Sum of Functions

The sum of functions is a way to combine two functions into a single new function. When given two functions, say \( f(x) = x - 3 \) and \( g(x) = x^2 \), the sum of these functions is represented as \((f+g)(x)\). To find this, you simply add the expressions of the functions:

• Combine like terms: Start by adding each part of \(f(x)\) and \(g(x)\), giving you \( (f+g)(x) = (x - 3) + x^2 \).
• Simplify: Reorder and combine the terms to get \( x^2 + x - 3 \).

This new function, \( x^2 + x - 3 \), is a polynomial function. Being a polynomial means its domain is all real numbers, \( \mathbb{R} \). You can choose any real number to substitute into this function and it will work perfectly every time.

Difference of Functions

Finding the difference of functions is similar to finding the sum, but involves subtracting one function from another. For the functions \( f(x) = x - 3 \) and \( g(x) = x^2 \), the difference is expressed as \( (f-g)(x) \). Here's how you do it:

• Subtract the functions: Write down \( (f-g)(x) = (x - 3) - x^2 \).
• Reorder and combine: Simplify this to put the polynomial in standard form: \( -x^2 + x - 3 \).

This expression, \( -x^2 + x - 3 \), is also a polynomial. Like with the sum, the domain is all real numbers, \( \mathbb{R} \). Polynomials have no restrictions on input values, which makes them quite flexible.

Product of Functions

The product of functions involves multiplying two functions together to get a new function. For \( f(x) = x - 3 \) and \( g(x) = x^2 \), the product is denoted as \( (fg)(x) \). Here's the process:

• Multiply the expressions: Take the expressions of \(f(x)\) and \( g(x)\), and multiply them: \( (fg)(x) = (x - 3) \cdot x^2 \).
• Distribute: Multiply each term in \( x^2 \) by each term in \( x - 3 \).
• Simplify: This results in \( x^3 - 3x^2 \), putting it in standard polynomial form.

Just like the sum and difference, this product function, \( x^3 - 3x^2 \), is a polynomial, so it shares the same domain: all real numbers, \( \mathbb{R} \).

Quotient of Functions

The quotient of functions is found by dividing one function by another, resulting in a new function. For the functions \( f(x) = x - 3 \) and \( g(x) = x^2 \), the quotient is represented as \( (f/g)(x) \):

• Set up the division: Write the quotient as \( \frac{f(x)}{g(x)} = \frac{x - 3}{x^2} \).
• Determine domain restrictions: The division is only valid when the denominator \( g(x) \) is not zero. So, solve \( x^2 = 0 \) for \( x \), which gives \( x = 0 \). This means you have to exclude \( x = 0 \) from the domain.
• Define the domain: The domain for \( \frac{x - 3}{x^2} \) becomes all real numbers except zero, \( \mathbb{R} \setminus \{0\} \).

When working with quotients, always check which values might make the denominator zero, as these need to be excluded from the domain.