Problem 3 - Thomas Calculus

Question

In Exercises \(1-6,\) find the domain and range of each function. $$ F(x)=\sqrt{5 x+10} $$

Step-by-Step Solution

Answer

Domain: \([-2, \infty)\), Range: \([0, \infty)\).

Step 1: Identify the Function Type

The given function is a square root function, which takes the form \( F(x) = \sqrt{5x + 10} \). Square root functions are defined only for values where the expression inside the square root is greater than or equal to 0. This is because the square root of a negative number is not a real number.

Step 2: Determine the Domain

Set the expression inside the square root (\( 5x + 10 \)) to be greater than or equal to 0: \ \[ 5x + 10 \geq 0 \]. \ Solve for \( x \): \ Subtract 10 from both sides: \ \[ 5x \geq -10 \] \ Divide both sides by 5: \ \[ x \geq -2 \]. \ Therefore, the domain of \( F(x) \) is \( x \geq -2 \), or in interval notation, \([-2, \infty)\).

Step 3: Determine the Range

The output of the function \( F(x) = \sqrt{5x + 10} \) will be limited by the square root, which outputs non-negative values. Thus, the smallest value \( F(x) \) can take is 0, when \( x = -2 \) (i.e., when \( 5(-2) + 10 = 0 \)). As \( x \) increases, \( F(x) \) increases without bound. Therefore, the range is \([0, \infty)\).

Key Concepts

Domain and Range

The domain and range of a function refer to the set of possible inputs (domain) and outputs (range) that a function can have. When dealing with functions, especially in calculus, understanding these aspects is crucial for analyzing the behavior of functions.**Domain**:- The domain is the complete set of all possible values of the independent variable, usually represented by \( x \), for which the function is defined.- For example, in the function \( F(x) = \sqrt{5x+10} \), the domain dictates that the values put into the function must lead to real numbers after calculation.- Because a square root function cannot take a negative number (in real numbers), the expression inside the square root needs to be greater than or equal to 0.**Range**:- Conversely, the range is the set of all possible output values, or \( F(x) \), that result from replacing \( x \) in the function with numbers from the domain.- For \( F(x) = \sqrt{5x+10}\), the smallest value the function can output is 0, which occurs when \( x = -2 \).- As \( x \) increases, the function output also increases without bound, leading to a range from 0 to infinity, or \([0, \, \infty)\).

Square Root Function

A square root function is a type of radical function where the expression inside the square root can drastically alter the function's behavior. Possible values for the argument must result in a non-negative number inside the radical because the square root of a negative number isn’t defined in the realm of real numbers. Key characteristics of square root functions:- **Shape**: They typically have a half-parabola shape, starting from a defined point and stretching horizontally with increasing values of \( x \).- **Dominance of the radicand**: The expression inside the root—called the radicand—determines where the graph of the function starts, ends, or extends to.- **Example in context**: For \( F(x) = \sqrt{5x + 10} \), the function starts at \( x = -2 \). As \( x \) increases, \( \sqrt{5x+10} \) also grows.Notably, these functions reveal important limits for inputs and outputs, which directly tie into the domain and range considerations.

Interval Notation

Interval notation is a way of expressing subsets of the real numbers, which is particularly useful for defining domains and ranges clearly.When using interval notation:- **Square brackets \([\, ]\)** indicate that an endpoint is included (closed interval).- **Parentheses \((\, )\)** indicate that an endpoint is not included (open interval).**Examples**: - The interval \([-2, \, \infty)\) includes all numbers from \(-2\) to infinity, including \(-2\) but not infinity, which is why the bracket is used to include \(-2\) and a parenthesis for infinity.- A closed interval such as \([0, 5]\) indicates all real numbers from 0 to 5, including both endpoints. Using such notation facilitates mathematical communication, helping people understand quickly the exact set of numbers being referred to and used in mathematical functions and problems.