Problem 32 - College Algebra Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=3 x-4, g(x)=x+2$$

Step-by-Step Solution

Answer

After performing the operations, \(f+g = 4x - 2\) and \(f-g = 2x - 6\), with both having domain as all the real numbers. For \(fg = 3x^2 + 2x - 8\), again the domain is all real numbers. For the division \(\frac{f}{g} = \frac{3x - 4}{x + 2}\), the domain is all real numbers except -2 because at \(x = -2\), the denominator equals to zero and will make the function undefined.

Step 1: Function Addition (f + g)

Add function f(x) and g(x) together. \(f(x) + g(x) = (3x - 4) + (x + 2) = 4x - 2 \). The domain is all real numbers since there isn't any value that makes this function undefined.

Step 2: Function Subtraction (f - g)

Subtract function g(x) from f(x). \(f(x) - g(x) = (3x - 4) - (x + 2) = 2x - 6 \). The domain is all real numbers because there is no value making this function undefined.

Step 3: Function Multiplication (fg)

Multiply function f(x) by g(x). \(f(x) \cdot g(x) = (3x - 4) \cdot (x + 2) = 3x^2 + 2x - 8 \). The domain is all real numbers because the function is valid for any real number.

Step 4: Function Division (\(\frac{f}{g}\))

Determine \(\frac{f(x)}{g(x)}\). \(\frac{f(x)}{g(x)} = \frac{3x - 4}{x + 2}\). The domain is all real numbers except -2 since the function is undefined at \(x = -2\), where the denominator equals to zero.

Key Concepts

Function Addition

Adding two functions together is a straightforward process. Think of it as combining elements from two separate sets. When you add function \( f(x) = 3x - 4 \) and function \( g(x) = x + 2 \), you simply add the corresponding terms together.

• First, combine the \( x \) terms: \( 3x + x = 4x \).
• Next, combine the constant terms: \( -4 + 2 = -2 \).

Together, these create the resulting function: \( f(x) + g(x) = 4x - 2 \). The domain of this new function is all real numbers, meaning it works for any number you plug in. There's no restriction, thanks to the lack of variables in the denominator or under a square root.

Function Subtraction

Subtracting one function from another involves finding the difference between their terms. For \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \), subtract \( g(x) \) from \( f(x) \):

• Start by subtracting the \( x \) terms: \( 3x - x = 2x \).
• Then, subtract the constants: \( -4 - 2 = -6 \).

The resulting function is \( f(x) - g(x) = 2x - 6 \). Its domain is also all real numbers. Again, there are no restrictions since there's no division by zero or issues with roots.

Function Multiplication

When multiplying two functions, each term from one function is multiplied by every term in the other. The functions \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \) can be multiplied as follows:

• Multiply \( 3x \) by each term in \( g(x) \) to get \( 3x^2 + 6x \).
• Multiply \(-4\) by each term in \( g(x) \) to get \(-4x - 8\).
• Add these results to get the final function: \( 3x^2 + 2x - 8 \).

The domain for this function also includes all real numbers, as polynomial functions do not have restrictions like division or square roots that could limit their domain.

Function Division

Dividing one function by another is a bit trickier because we need to ensure the denominator does not equal zero. With functions \( f(x) = 3x - 4 \) and \( g(x) = x + 2 \), their division is:\[\frac{f(x)}{g(x)} = \frac{3x - 4}{x + 2}\]We need to determine when \( g(x) = 0 \) because division by zero is undefined.

• Solving \( x + 2 = 0 \) gives \( x = -2 \).

This means the domain excludes \( x = -2 \). Thus, the domain is all real numbers except \(-2\). Pay special attention to the denominator to find any domain restrictions.

Domain of a Function

The domain of a function refers to all the possible input values (usually \( x \)) the function can accept. For functions without denominators or roots, like linear polynomials, the domain usually includes all real numbers.However, when there's a division like \( \frac{f(x)}{g(x)} \), restrictions can occur:

• If \( g(x) = 0 \), the function is undefined at that point.

For example, in \( \frac{3x - 4}{x + 2} \), \( x \) cannot be \(-2\), ensuring you do not divide by zero. In any algebraic problem, always check if the functions involve division or roots, as these impact the domain.