Problem 29 - 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) \((f / g)(x)\) $$ f(x)=x^{5 / 2}-x^{3 / 2}, \quad g(x)=x^{1 / 2} $$

Step-by-Step Solution

Answer

The expressions are (a) \(x^{5/2} - x^{3/2} + x^{1/2}\), (b) \(x^{5/2} - x^{3/2} - x^{1/2}\), (c) \(x^3 - x^2\), (d) \(x^2 - x\) with domains of \(x \geq 0\) for (a) and (b), and \(x > 0\) for (c) and (d).

Step 1: Define (f + g)(x)

To find \((f + g)(x)\), we add the functions \(f(x)\) and \(g(x)\). This gives us:\[f(x) + g(x) = \left(x^{5/2} - x^{3/2}\right) + x^{1/2}\] Simplifying, \[(f+g)(x) = x^{5/2} - x^{3/2} + x^{1/2}\] The domain consists of all \(x\) such that both \(f(x)\) and \(g(x)\) are defined. Since they both have \(x\) raised to fractional powers, \(x\geq 0\).

Step 2: Define (f - g)(x)

To find \((f - g)(x)\), subtract \(g(x)\) from \(f(x)\):\[f(x) - g(x) = \left(x^{5/2} - x^{3/2}\right) - x^{1/2}\] Simplifying, \[(f-g)(x) = x^{5/2} - x^{3/2} - x^{1/2}\] The domain is \(x\geq 0\) as both functions must be defined.

Step 3: Define (fg)(x)

For the product \((f g)(x)\), multiply \(f(x)\) and \(g(x)\):\[f(x) \, g(x) = \left(x^{5/2} - x^{3/2}\right) x^{1/2}\] Expanding,\[(fg)(x) = x^{5/2 + 1/2} - x^{3/2 + 1/2} = x^{3} - x^{2}\] The domain is \(x\geq 0\) again due to the presence of fractional powers.

Step 4: Define (f / g)(x)

For the quotient \((f / g)(x)\), divide \(f(x)\) by \(g(x)\):\[(f/g)(x) = \frac{x^{5/2} - x^{3/2}}{x^{1/2}}\] Simplifying, by dividing each term by \(x^{1/2}\),\[(f/g)(x) = x^{2} - x^{1}\]The domain is \(x > 0\) since \(g(x) = x^{1/2}\) must be non-zero.

Key Concepts

Function Operations

Function operations involve combining two or more functions in various ways. Just like adding or multiplying numbers, we can apply these operations to functions: addition, subtraction, multiplication, and division.
To perform these operations, we simply apply the operation to the output values of the functions. For example, to add two functions \(f(x)\) and \(g(x)\):

• Add the output of \(f(x)\) to the output of \(g(x)\).
• This gives the function \((f+g)(x) = f(x) + g(x)\).

If we are subtracting, multiplying, or dividing functions, we follow the same logic:

• Subtraction: \((f-g)(x) = f(x) - g(x)\)
• Multiplication: \((fg)(x) = f(x) \cdot g(x)\)
• Division: \((f/g)(x) = \frac{f(x)}{g(x)}\)

Remember, each operation can change the domain of the resulting function, so it is crucial to check it!

Domain of a Function

The domain of a function is the set of all possible input values (or \(x\)-values) that make the function defined and result in a real output.
When dealing with operations of functions, determining the domain of the resulting function involves considering the domains of the individual functions involved:

• For addition and subtraction, the domain consists of \(x\)-values common to both functions' domains.
• In multiplication, ensure that \(x\) is in the domain of both functions to correctly define the product function.
• For division, it's crucial to exclude any \(x\)-values where the denominator function is zero.

For the functions \(f(x) = x^{5/2} - x^{3/2}\) and \(g(x) = x^{1/2}\), the common domain is \(x \geq 0\) because both functions involve square roots, which require non-negative \(x\). In the case of division, \(x > 0\) because dividing by zero is undefined.

Simplifying Expressions

Simplifying expressions is about rewriting them in a form that is easier to understand or work with. In mathematical terms, simplification often involves:

• Combining like terms.
• Reducing complex fractions.
• Performing arithmetic operations.

For instance, consider the functions from our exercise. When finding \((f+g)(x)\), we combined and simplified like terms: \[f(x) + g(x) = x^{5/2} - x^{3/2} + x^{1/2}\]In multiplication, we expanded the expression:\[f(x) \cdot g(x) = x^{5/2} \cdot x^{1/2} - x^{3/2} \cdot x^{1/2} = x^3 - x^2\]When simplifying, always aim to reach the simplest expression, remembering to respect the mathematical properties like distribution or commutativity.