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

Domain: \([0, \infty)\), Range: \((- fty, 1]\).

Step 1: Identify the Function Type

The function provided is \( f(x) = 1 - \sqrt{x} \). This is a square root function subtracted from 1.

Step 2: Determine the Domain

The square root function \( \sqrt{x} \) requires that \( x \geq 0 \) since the square root of a negative number is undefined in the set of real numbers. Therefore, the domain of \( f(x) \) is \( x \geq 0 \). In interval notation, this is \([0, \infty)\).

Step 3: Find the Range of the Root Function

The \( \sqrt{x} \) function has a range of \([0, \infty)\). This means for any \( x \geq 0 \), the value of \( \sqrt{x} \) ranges from 0 to \( \infty\).

Step 4: Determine the Range of the Original Function

Since \( f(x) = 1 - \sqrt{x} \), we take the range of \( \sqrt{x} \), which is \([0, \infty)\), and apply the transformation: subtract each value from 1. This transforms the minimum value (0) into 1-0 = 1 and pushes the maximum down towards \(-\infty\) as \( \sqrt{x} \) increases. Thus, the range of \( f(x) \) is \((-fty , 1]\).

Step 5: Conclusion on Domain and Range

The domain of the function \( f(x) = 1 - \sqrt{x} \) is \([0, \infty)\), and the range is \((-fty , 1]\).

Key Concepts

Square Root Functions

The concept of square root functions is essential in understanding how particular functions behave. A square root function usually takes the form of \( f(x) = \sqrt{x} \). This type of function grows gradually as \( x \) increases. It is important to note that square root functions are defined only for non-negative numbers.

• When the argument of the square root, \( x \), is less than zero, the function becomes undefined in the realm of real numbers.
• The domain of a standard square root function is \( x \geq 0 \), represented in interval notation as \([0, \infty)\).
• The range of \( \sqrt{x} \) covers non-negative real numbers, which means it is also \([0, \infty)\).

Understanding these domain and range limitations helps us grasp why the function \( f(x) = 1 - \sqrt{x} \) changes in its domain and range as transformations are applied.

Function Transformation

Function transformations involve shifting, stretching, or compressing a function in various ways. In the function \( f(x) = 1 - \sqrt{x} \), a simple transformation is applied: the square root function is subtracted from 1. This alteration affects both its appearance and range.

• Transformation by subtraction: The whole graph of the square root function is shifted downward by subtracting each \( y \)-value from a constant (1 in this case).
• Shifting effects: Because of this shift, the maximum value at the function's starting point (when \( x = 0 \)) changes from 0 to 1.
• As \( \sqrt{x} \) increases, the values become negative, moving towards \(-\infty\).

This horizontal stretching or shifting allows us to determine a new range for \( f(x) = 1 - \sqrt{x} \), resulting in \((-\infty, 1]\). Understanding how these transformations work can give insight into manipulating functions for desired properties.

Real Numbers

Real numbers are a cornerstone of understanding functions like square root functions and their transformations. They form the extended set of values where many mathematical operations take place.

• Real numbers include all the rational numbers, such as integers and fractions, as well as all the irrational numbers, which cannot be expressed as fractions, such as \( \sqrt{2} \).
• In the context of square roots, real numbers determine the existence of square roots. For instance, \( \sqrt{9} = 3 \) is valid because 9 is a non-negative real number.
• In functions, we often restrict operations to real numbers to avoid undefined expressions, such as trying to find \( \sqrt{-4} \), which isn't possible within the real number system.

Real numbers offer a framework to work with domains and ranges effectively, ensuring that functions behave as expected. It is by staying within real numbers that we confirm the domain of \( f(x) = 1 - \sqrt{x} \) as all non-negative numbers, \([0, \infty)\), to avoid undefined results.