Problem 18 - 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) \((f / g)(x)\) $$ f(x)=\sqrt{1-x}, \quad g(x)=x^{3} $$

Step-by-Step Solution

Answer

(a) \(\sqrt{1-x} + x^3\), \(x \leq 1\); (b) \(\sqrt{1-x} - x^3\), \(x \leq 1\); (c) \(\sqrt{1-x} \cdot x^3\), \(x \leq 1\); (d) \(\frac{\sqrt{1-x}}{x^3}\), \(x < 1, x \neq 0\).

Step 1: Understand the Functions

The given functions are \(f(x) = \sqrt{1-x}\) and \(g(x) = x^3\). \(f(x)\) is a square root function, and \(g(x)\) is a cubic function.

Step 2: Find the Formula for (f+g)(x)

The expression \((f+g)(x)\) is defined as the sum of \(f(x)\) and \(g(x)\). Therefore,\[(f+g)(x) = f(x) + g(x) = \sqrt{1-x} + x^3\]To find the domain, both \(\sqrt{1-x}\) and \(x^3\) must be defined, which requires \(1-x \geq 0\). Thus, the domain is \(x \leq 1\).

Step 3: Find the Formula for (f-g)(x)

The expression \((f-g)(x)\) is defined as \(f(x) - g(x)\). Therefore,\[(f-g)(x) = f(x) - g(x) = \sqrt{1-x} - x^3\]The domain remains the same as \(f(x)\)'s, which is \(x \leq 1\).

Step 4: Find the Formula for (f*g)(x)

The expression \((f \cdot g)(x)\) is defined as \(f(x) \cdot g(x)\). Therefore,\[(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{1-x} \cdot x^3\]The domain is determined by \(f(x)\), so \(x \leq 1\).

Step 5: Find the Formula for (f/g)(x)

The expression \((f/g)(x)\) is defined as \(f(x)/g(x)\). Therefore,\[(f/g)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{1-x}}{x^3}\]This expression is only defined when \(g(x) eq 0\) and \(1-x \geq 0\). Thus, the domain is \(x < 1\) and \(x eq 0\) (to avoid division by zero).

Key Concepts

Domain

When we talk about the domain of a function, we are referring to the set of all possible input values (usually denoted as 'x') that can be used in the function without causing any mathematical problems, like division by zero or taking the square root of a negative number.

For any mathematical operation involving functions, getting the domain right is crucial. Here's how you can think about it:

• Consider each part of the function separately and identify their individual domains.
• Combine these considerations to find the overall domain of the operation you're analyzing.

This means for operations like addition, subtraction, multiplication, and division, we need to ensure each part is defined at any given point.

For example, in operations involving the functions \(f(x) = \sqrt{1-x}\) and \(g(x) = x^3\), the domain for operations like addition or subtraction would be the intersection of their valid domains.

In our specific exercise, the square root function \(f(x) = \sqrt{1-x}\) requires \(1-x \geq 0\), so \(x \leq 1\). While the cubic function \(g(x) = x^3\) is defined for all real numbers. Pay special attention during division operations, because in \((f/g)(x)\), \(x = 0\) is excluded to avoid division by zero, further limiting the domain as \(x < 1\) and \(x eq 0\).

Square Root Function

The square root function is a basic yet essential function in mathematics, defined generally as \(f(x) = \sqrt{x}\). It only accepts non-negative inputs because the square root of a negative number results in an imaginary number not definable on the real number line.

In our problem's context, the function \(f(x) = \sqrt{1-x}\) incorporates a constraint within the root. The expression inside the square root, \(1-x\), must remain non-negative. This results in the condition \(1-x \geq 0\), meaning the domain of this function is all \(x\) such that \(x \leq 1\).

• Understanding this is key for operations like addition or multiplication with other functions, as it sets limits on which values of \(x\) are permissible.
• Visualizing the graph can also help! The graph of \(\sqrt{1-x}\) is a curve that starts from the point \((1,0)\) and moves gracefully towards \((0, 1)\) approaching infinity along the y-axis.

Remember, the square root function impacts the feasibility of operations because it dictates the primary domain restrictions.

Cubic Function

The cubic function, expressed as \(g(x) = x^3\), is a fascinating type of polynomial function. It's a continuous curve defined for all real numbers, making its domain the entire real number line:

• It doesn't have any restrictions or constraints, unlike root functions.
• This makes it an interesting case for function operations since its unrestricted domain means it won't usually limit the resulting function's domain.

On a graph, the cubic function starts from negative infinity, passes through the origin, and heads toward positive infinity.

The cubic function showcases unique characteristics:

• It's symmetric with respect to the origin, meaning \(g(-x) = -g(x)\).
• Its shape is distinct, starting with negative values for \(x < 0\), crossing zero at \(x = 0\), and increasing rapidly for \(x > 0\).

Understanding these properties is essential, especially when engaging in operations involving cubic and other types of functions, such as determining their combined effects or explaining any shifts in domain or range related to mixed operations.