Problem 16 - Precalculus with Limits

Question

In Exercises 9-16, find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((fg)(x)\), and (d) \((f/g)(x). What is the domain of \)f/g\(? \)f(x) = \frac{x}{x+1}\(, \)g(x) = x^3$

Step-by-Step Solution

Answer

(a) (f+g)(x)=\(\frac{x}{x+1} + x^3\), (b) (f-g)(x)=\(\frac{x}{x+1} - x^3\), (c) (fg)(x) = \(\frac{x^4}{x+1}\), (d) (f/g)(x) = \(\frac{x}{x^4 + x^3}\). The domain of (f/g)(x) is all real numbers except 0.

Step 1: Finding the Sum of f(x) and g(x)

The sum of two functions (f+g)(x) is the sum of their expressions. So (f+g)(x) = \(\frac{x}{x+1} + x^3.\)

Step 2: Finding the Difference of f(x) and g(x)

The difference of two functions (f-g)(x) is the difference of their expressions. So (f-g)(x) = \(\frac{x}{x+1} - x^3.\)

Step 3: Finding the Product of f(x) and g(x)

The product of two functions (fg)(x) is the product of their expressions. So (fg)(x) = \(\frac{x}{x+1} . x^3 = \frac{x^4}{x+1}.\)

Step 4: Finding the Quotient of f(x) and g(x) and its Domain

The quotient of two functions (f/g)(x) is the quotient of their expressions, as long as g(x) ≠ 0. So (f/g)(x) = \(\frac{\frac{x}{x+1}}{x^3} = \frac{x}{x^4 + x^3}\). g(x) = \(x^3\), so g(x) ≠ 0 when x ≠ 0. Therefore, the domain of f/g is all real numbers except 0.

Key Concepts

Sum of Functions

When you add two functions together, you perform what's called the sum of functions operation. It involves adding their respective outputs. If you have the functions \( f(x) \) and \( g(x) \), their sum is expressed as \((f+g)(x) = f(x) + g(x)\).
For example, with \( f(x) = \frac{x}{x+1} \) and \( g(x) = x^3 \), their sum \((f+g)(x)\) becomes \( \frac{x}{x+1} + x^3 \).
This operation allows you to combine two different mathematical rules into a new rule, which also combines their values for any input \( x \).

• Make sure both functions are defined for the same domain!

Difference of Functions

Subtracting one function from another is known as the difference of functions. It is simply the subtraction of the outputs of two functions. Given \( f(x) \) and \( g(x) \), the difference is \((f-g)(x) = f(x) - g(x)\).
For the functions \( f(x) = \frac{x}{x+1} \) and \( g(x) = x^3 \), their difference \((f-g)(x)\) is \( \frac{x}{x+1} - x^3 \).
Understanding this concept helps in manipulating and simplifying functions to see how they interact differently.

• Perform subtraction carefully to avoid mistakes.

Product of Functions

The product of functions involves multiplying two functions together. Basically, you multiply their respective outputs. If you have \( f(x) \) and \( g(x) \), their product \((fg)(x)\) is written as \( f(x) \cdot g(x) \).
In our example \( f(x) = \frac{x}{x+1} \) and \( g(x) = x^3 \), the product \((fg)(x)\) becomes \( \frac{x^4}{x+1} \).
This operation helps us understand how two functions work together to change the resulting output significantly.

• Remember to consider the domain restrictions after multiplication.

Quotient of Functions

The quotient of functions is where you divide one function by another. Expressed as \((f/g)(x) = \frac{f(x)}{g(x)}\), this operation can only be done if the denominator function \( g(x) \) is not zero.
Using our functions \( f(x) = \frac{x}{x+1} \) and \( g(x) = x^3 \), the quotient \((f/g)(x)\) results in \( \frac{x}{x^4 + x^3} \).
It's important to make sure the function in the denominator doesn't equal zero at any point in the domain of the composite function.

• Always check \( g(x) eq 0 \) to determine valid input values!

Domain of a Function

The domain of a function is the set of possible input values (\( x \)-values) for which the function is defined.
For the quotient \( (f/g)(x) = \frac{x}{x^4 + x^3} \), you need to find where both \( f(x) \) and \( g(x) \) are defined, ensuring \( g(x) eq 0 \).
In this case, since \( g(x) = x^3 \), it becomes zero when \( x = 0 \). Thus, the function \( f/g \) is defined for all real numbers except \( x = 0 \).

• Consider both numerator and denominator for domain restrictions.
• Finding the domain is crucial for understanding where a function can and cannot exist.