Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For each function in Problems 22 through 27, determine the largest possible domain. (a) \(f(x)=\frac{1}{5 x+10}\) (b) \(g(x)=\sqrt{5 x+10}\)

Step-by-Step Solution

Answer

The domain of \(f(x)=\frac{1}{5 x+10}\) is all real numbers except -2 and the domain of \(g(x)=\sqrt{5 x+10}\) is all real numbers equal to or greater than -2.

Step 1: Determine the domain of \(f(x)=\frac{1}{5 x+10}\)

To find the domain, simply set the denominator of the fraction equal to zero and then find the values of x that make this true, since those are the values where the function is undefined. So, \(5 x+10 = 0\). Solving for x, it is found that x = -2. Therefore, the domain of the function is all real numbers except -2.

Step 2: Determine the domain of \(g(x)=\sqrt{5 x+10}\)

The square root function is defined as long as the expression inside the square root is non-negative (greater than or equal to zero). To find the domain, you'll once again set the expression inside the square root equal to zero, \(5 x+10 = 0\), and solve for x. It is found that x = -2. This is the value at which the function starts being defined, so the domain is all x values that are greater than or equal to -2.

Key Concepts

Rational Functions

Rational functions are a crucial concept in algebra. They are essentially functions that can be written as a fraction, where both the numerator and the denominator are polynomials. The general form for a rational function looks like this: \[ f(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomial expressions and \( Q(x) eq 0 \).

• Finding the Domain: The domain of a rational function is the set of all real numbers except for the values that make the denominator zero. To find these values, you simply set the denominator equal to zero and solve for \( x \). This step is key because a division by zero is undefined in mathematics.
• Example: Consider \( f(x) = \frac{1}{5x + 10} \). Here, the denominator \( 5x + 10 \) can be zero when \( 5x + 10 = 0 \). Solving for \( x \) yields \( x = -2 \). Thus, the domain is all real numbers except \( x = -2 \).

Understanding rational functions helps us predict or eliminate values that could make the function undefined. These insights are essential for graphing rational functions or solving real-world problems involving rates and proportions.

Square Root Functions

Square root functions involve taking the square root of an expression, such as \( g(x) = \sqrt{5x + 10} \). Such functions are interesting because they inherently limit the possible values for \( x \), which impacts their domain. The domain of a square root function occurs when the expression inside the root is non-negative, as square roots of negative numbers are not defined in the set of real numbers. This means we need the expression beneath the square root to be zero or greater.

• Determining the Domain: To find this, you set the expression inside the square root \( \sqrt{5x + 10} \geq 0 \). Solving \( 5x + 10 \geq 0 \) gives \( x \geq -2 \).
• Practical Applications: Square root functions often model scenarios where quantities cannot be negative, such as calculating lengths, areas, or statistics.

Thus, the domain of \( g(x) = \sqrt{5x + 10} \) comprises all \( x \) values greater than or equal to \(-2\). Recognizing the constraints placed by square roots is invaluable in both solving mathematical problems and understanding certain real-world limitations.

Mathematical Expressions

Mathematical expressions form the building blocks of algebra, involving numbers, variables, and operations. It's essential to understand how different expressions work individually and interact within functions like rational or square root functions. A mathematical expression might include several components: numbers, variables like \( x \), arithmetic operations like addition or multiplication, or more complex aspects like exponents and radicals.

• Operations with Expressions: When dealing with these expressions, remember that the rules of arithmetic still apply. You can add, subtract, multiply, divide, and simplify these expressions just as you would with regular numbers.
• Expressions in Different Functions: For rational functions, the expression is the polynomial in the numerator and denominator; for square root functions, it’s the content under the root.

For clarity, expressions are typically manipulated to determine the domain of functions or simplify equations. In our examples, we examined expressions within the context of rational and square root functions to understand where they are defined or undefined, giving us the domain. Mastery of expressions and their simplifications leads to success in more advanced mathematics.