Problem 2 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The domain is \([0, \infty)\) and the range is \((-\infty, 1]\).

Step 1: Identify the Domain of the Function

The function is defined as \( f(x) = 1 - \sqrt{x} \). The square root function \( \sqrt{x} \) is defined only for non-negative values of \( x \), meaning \( x \geq 0 \). Therefore, the domain of the function is \([0, \infty)\).

Step 2: Determine the Range of the Function

Given the domain \([0, \infty)\), consider the behavior of \( f(x) = 1 - \sqrt{x} \). When \( x = 0 \), \( f(0) = 1 \). As \( x \) increases towards infinity, \( \sqrt{x} \) also increases towards infinity, making \( f(x) \) approach \(-\infty\). Thus, the range of the function is \((-\infty, 1]\).

Key Concepts

Functions

A function is a mathematical relationship between two sets of numbers, where each input is assigned one and only one output. This relationship is often represented by an equation like \( f(x) = 1 - \sqrt{x} \). When you think of functions, picture them as machines that take an input, perform a specific operation, and give an output:

• The input value is referred to as "\( x \)" or the "independent variable".
• The output, on the other hand, is the function's value, often written as "\( f(x) \)" or "\( y \)".

In the exercise provided, we examined the function \( f(x) = 1 - \sqrt{x} \), where the operation involves extracting the square root of "\( x \)" and subtracting it from 1. Understanding this structure is crucial for determining both the domain and the range of the function.

Square Root

The square root function, represented by the symbol \( \sqrt{x} \), plays a significant role in defining the domain of a given function. By definition, the square root is the value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\).In terms of domain, the square root function poses a limitation:

• It is only defined for non-negative numbers, \( x \geq 0 \), because the square root of a negative number isn't a real number (in basic real-number mathematics).
• This limitation directly affects the function \( f(x) = 1 - \sqrt{x} \), which can only take non-negative inputs.

Therefore, knowing when the square root function appears in an equation helps us establish that the domain is typically limited to \([0, \infty)\), ensuring all values inside the function are conceivable and valid. Identifying these boundaries is a critical step in solving any function-related problems.

Infinity

Infinity is a concept used to describe something without any limit, often appearing in mathematics as a way to express unboundedness. In the context of function domains and ranges, infinity becomes particularly relevant.In the exercise, infinity is used to define both the domain and the range of the function:

• The domain \([0, \infty)\) suggests that while the function starts at a bound \( x = 0 \), it is not limited at the other end, and can grow indefinitely along the positive \( x \)-axis.
• The range \((-\infty, 1]\) indicates that while the minimum output value starts at \( f(0) = 1 \), the function can decrease without bound as \( x \) increases, approaching negative infinity.

These uses of infinity illustrate how mathematicians deal with concepts that extend beyond finite comprehension, allowing us to work with functions that reach extremely large positive or negative values. Understanding this concept helps with grasping both the behavior and the solutions of complex equations.