Problem 5 - Precalculus

Question

Find \((a)(f+g)(x),(b)(f-g)(x)\) (c) \((f g)(x),\) and \((d)(f g)(x) .\) What is the domain of \(f g ?\) $$f(x)=x+2, \quad g(x)=x-2$$

Step-by-Step Solution

Answer

(a) \((f+g)(x)\) = 2x, (b) \((f-g)(x)\) = 4, (c) \((fg)(x)\) = \(x^2 - 4\), (d) \((f/g)(x)\) = \((x + 2) / (x - 2) \) with a domain of \(\{x : x ≠ 2\}\). The domain of \((fg)\) is the set of all real numbers.

Step 1: Function Addition

The sum of the functions \(f(x)\) and \(g(x)\) is calculated by adding them together: \((f+g)(x) = f(x) + g(x) = (x + 2) + (x - 2) = 2x.\

Step 2: Function Subtraction

The difference of the functions \(f(x)\) and \(g(x)\) is calculated by subtracting \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = (x + 2) - (x - 2) = 4.

Step 3: Function Multiplication

The product of the functions \(f(x)\) and \(g(x)\) is found by multiplying them together: \((fg)(x) = f(x) * g(x) = (x + 2) * (x - 2) = x^2 - 4.

Step 4: Function Division

The quotient of \(f(x)\) divided by \(g(x)\) is calculated as \((f/g)(x) = f(x) / g(x) = (x + 2) / (x - 2)\. But we need to ensure that the denominator \(g(x)\) is not equal to zero, because division by zero is undefined. Therefore, the domain for this function is \(\{x : x ≠ 2\}\.

Step 5: Determine the Domain of Function Multiplication

The domain for the product of \(f(x)\) and \(g(x)\) is the set of all real numbers, because the function produced, \(x^2 - 4\), is defined for all real values of \(x\).

Key Concepts

Function Addition

Function addition involves combining two functions into a single function by adding their outputs. Let's consider the functions given:

• \(f(x) = x + 2\)
• \(g(x) = x - 2\)

To find \((f+g)(x)\), simply add the two functions together:

• \( (f+g)(x) = f(x) + g(x) = (x + 2) + (x - 2) \)
• This simplifies to \(2x\).

Therefore, the new function \((f+g)(x)\) is \(2x\), which is a linear function. The process of function addition is straightforward—just remember you are adding the complete expressions together before simplifying.

Function Subtraction

Function subtraction is similar to function addition, but instead of adding, you subtract one function from the other. Using the same functions \(f(x) = x + 2\) and \(g(x) = x - 2\):

• Subtract \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = (x + 2) - (x - 2)\)
• When you perform the subtraction, make sure to distribute any negative signs correctly. Here, subtracting \((x - 2)\) changes the signs.
• After simplifying, you get \(4\).

The result is a constant function \((f-g)(x) = 4\). Always simplify carefully to ensure the correct result.

Function Multiplication

Function multiplication involves multiplying the outputs of two functions to produce a new function. Here's how to multiply \(f(x) = x + 2\) and \(g(x) = x - 2\):

• mulitply the expressions: \((fg)(x) = f(x) \times g(x) = (x + 2) \times (x - 2)\)

Use the distributive property:

• \((x + 2)(x - 2) = x^2 - 2x + 2x - 4\)
• The middle terms \(-2x\) and \(+2x\) cancel each other out.

Simplifying, you get:

• \((fg)(x) = x^2 - 4\).

The result \(x^2 - 4\) is a quadratic function, which can be factored further if necessary.

Domain of a Function

The domain of a function is the set of all possible input values (\(x\)-values) that the function can accept. Generally, we consider restrictions such as division by zero or taking the square root of negative numbers.

• For function multiplication, the domain is typically the intersection of the domains of the individual functions.
• In our case, the functions \(f(x)\) and \(g(x)\) are defined for all real numbers, so by multiplying them, \((fg)(x) = x^2 - 4\), the resulting function also has a domain of all real numbers.
• There are no values of \(x\) that produce undefined results for \((fg)(x)\) since no division by zero or square roots are involved.

For function division, however, we must exclude any value for \(x\) that makes the denominator zero. Therefore, when dealing with division operations, always check the domain!