Problem 23 - College Algebra with Modeling and Visualization

Question

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=x^{2}-1, \quad g(x)=|x+1| $$

Step-by-Step Solution

Answer

(a) \((f+g)(x) = x^2 - 1 + |x+1|\); all reals. (b) \((f-g)(x) = x^2 - 1 - |x+1|\); all reals. (c) \((fg)(x) = (x^2 - 1)|x+1|\); all reals. (d) \((f/g)(x) = \frac{x^2 - 1}{|x+1|}\); \(x \neq -1\).

Step 1: Add the Functions

To find \((f+g)(x)\), add the functions \(f(x) = x^2 - 1\) and \(g(x) = |x+1|\):\[(f+g)(x) = f(x) + g(x) = (x^2 - 1) + |x+1|\].The domain is all real numbers since both \(x^2 - 1\) and \(|x+1|\) are defined for any \(x\).

Step 2: Subtract the Functions

To find \((f-g)(x)\), subtract \(g(x) = |x+1|\) from \(f(x) = x^2 - 1\):\[(f-g)(x) = f(x) - g(x) = (x^2 - 1) - |x+1|\].The domain remains all real numbers for the same reason as in Step 1.

Step 3: Multiply the Functions

To find \((fg)(x)\), multiply \(f(x) = x^2 - 1\) by \(g(x) = |x+1|\):\[(fg)(x) = f(x) \, g(x) = (x^2 - 1) |x+1|\].The domain is all real numbers, as the product of these functions is defined for any \(x\).

Step 4: Divide the Functions and Determine Domain

To find \((f/g)(x)\), divide \(f(x) = x^2 - 1\) by \(g(x) = |x+1|\):\[(f/g)(x) = \frac{x^2 - 1}{|x+1|}\].The domain is all real numbers except \(x = -1\) because \(|x+1|\) equals zero at \(x = -1\), making the expression undefined.

Key Concepts

Function Operations

In the world of algebraic functions, function operations are a set of methods that allow us to create new functions by combining existing ones through addition, subtraction, multiplication, and division. Imagine you have two functions, \( f(x) \) and \( g(x) \). You can combine these functions in several ways:

• Addition: The sum of \( f(x) \) and \( g(x) \) is expressed as \((f+g)(x) = f(x) + g(x)\). This operation combines the two functions by adding their outputs for the same input \(x\).
• Subtraction: The difference between \( f(x) \) and \( g(x) \) is given by \((f-g)(x) = f(x) - g(x)\). Here, the outputs of \( g(x) \) are subtracted from \( f(x) \).
• Multiplication: To multiply the functions, you calculate \((fg)(x) = f(x) \cdot g(x)\). This product is formed by multiplying the outputs of both functions for each input \(x\).
• Division: The division of \( f(x) \) by \( g(x) \) is \((f/g)(x) = \frac{f(x)}{g(x)}\). Keep in mind that division requires special attention, as you cannot divide by zero.

These operations can be done on any functions, creating new relationships and expanding the ways you can analyze mathematical problems.

Domain of a Function

The domain of a function is a critical concept to understand because it tells you all possible input values \(x\) for which the function is defined. Think of it as the set of all "legal" \(x\)-values you can use without causing mathematical mishaps.For most arithmetic operations, like addition and subtraction, the domain is typically all real numbers. However, special cases occur in functions involving division and square roots:

• Division: When dividing \(f(x)\) by \(g(x)\), you must avoid any \(x\) that makes \(g(x) = 0\), as division by zero is undefined.
• Square Roots and Even Roots: For functions containing square roots, like \(\sqrt{x}\), the domain requires a non-negative radicand. This means \(x \geq 0\).

In our exercise, for example, \(f(x) = x^2 - 1\) and \(g(x) = |x+1|\) have domains of all real numbers because these functions are defined for any \(x\). However, when finding \((f/g)(x)\), we must exclude \(x = -1\) since \(|x+1|\) becomes zero, making division impossible.

Composite Functions

Composite functions arise when the output of one function becomes the input of another, forming a chain of functions. If you have two functions, \( f(x) \) and \( g(x) \), you can create a new function called a composite, denoted \((f \circ g)(x)\). This translates into \(f(g(x))\).To grasp this idea, consider:

• The Process: First, input a value of \(x\) into \(g(x)\), and calculate the result. Then, use this result as the input for \(f(x)\). The outcome from this last step is the value of the composite function.
• The Domain: Determining the domain of a composite function requires considering both functions involved. You must ensure that the \(g(x)\) values fit inside the domain of \(f(x)\).

Composite functions can be a bit more complex than straightforward function operations. However, they empower you to explore more intricate relationships between variables. As you practice, remember to evaluate the domain carefully to avoid undefined results.