Problem 86 - Intermediate Algebra

Question

Identify the domain and then graph each function. $$ f(x)=\sqrt{x}-2 $$

Step-by-Step Solution

Answer

The domain is \([0, \infty)\) and the graph is a downward shift of \( y=\sqrt{x} \).

Step 1: Understanding the Function

The given function is \( f(x) = \sqrt{x} - 2 \). It consists of a square root function (\( \sqrt{x} \)) shifted down by 2 units. We need to determine the domain of this function.

Step 2: Determining the Domain

The function includes a square root, which means \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( f(x) = \sqrt{x} - 2 \) is all non-negative real numbers, expressed in interval notation as \([0, \infty)\).

Step 3: Setting Up the Graph

To graph \( f(x) = \sqrt{x} - 2 \), we start with the basic graph of \( y = \sqrt{x} \) and then shift it vertically downward by 2 units.

Step 4: Graphing the Function

Plot the graph of \( f(x) = \sqrt{x} - 2 \) by choosing a few values of \( x \) within the domain. For example:- If \( x = 0 \), then \( f(0) = \sqrt{0} - 2 = -2 \).- If \( x = 1 \), then \( f(1) = \sqrt{1} - 2 = -1 \).- If \( x = 4 \), then \( f(4) = \sqrt{4} - 2 = 0 \).Plot these points and draw a smooth curve starting at \((0, -2)\) and moving upwards to the right.

Key Concepts

Square Root Function

The square root function is a type of function where the input (\( x \)) is placed inside a square root symbol. It is generally written as \( f(x) = \sqrt{x} \). This function is fundamental in algebra and is characterized by several unique properties.

First and foremost, the square root function is only defined for non-negative numbers. This means that \( x \) has to be positive or zero for the square root to provide a real number output. The reason is simple: negative numbers do not have real square roots within the realm of real numbers.

Key features of the square root function include:

• Non-negative domain: \( x \geq 0 \)
• Begins at the origin (0,0) if it is \( y = \sqrt{x} \)
• It increases at a decreasing rate as \( x \) grows

A square root function can undergo transformations, such as shifts or stretches, which change its graph's position or shape. For instance, in the function \( f(x) = \sqrt{x} - 2 \), the graph is shifted downward by 2 units.

Interval Notation

Interval notation is a way of representing a set of numbers (usually as the domain or range of a function) in a concise form. It allows us to define a continuum of values in a very simple manner.

When we express the set of input values (\( x \)) for a function, known as the domain, interval notation proves quite useful. For the function \( f(x) = \sqrt{x} - 2 \), we determined that \( x\) must be non-negative. Thus, the domain is all real numbers starting from 0 to infinity, which is written as \([0, \infty)\).

Here's a breakdown of the components of interval notation:

• **Brackets [ ]**: A square bracket \([\ or \ ]\) denotes that the endpoint is included in the interval. For example, \([0, 5]\) includes both 0 and 5.
• **Parentheses ( )**: A parenthesis \((\ or \ )\) indicates that the endpoint is not included. For instance, \((0, 5)\) does not include 0 or 5.
• **Infinity (∞)**: The symbol \(\infty\) is used to show that the interval extends beyond any finite number. It always comes with a parenthesis because infinity is not a specific number and cannot be reached.

Understanding interval notation is crucial for communicating the scope of a function's domain quickly and efficiently.

Graphing Functions

Graphing is a visual way of displaying the behavior or values of a function. With the graph of a function like \( f(x) = \sqrt{x} - 2 \), you can easily see how the function behaves as \( x \) changes.

To begin graphing, it's often necessary to start by recognizing the basic form of the function, like \( y = \sqrt{x} \) as the square root function's graph. The next step involves understanding any transformations that have been applied, such as shifts.

In our example, the graph of \( y = \sqrt{x} \) is shifted downward by 2 units. To graph it:

• Choose a variety of \( x \) values within the domain, in this case, \( x \geq 0 \).
• Calculate the corresponding \( y \) values using the function \( f(x) = \sqrt{x} - 2 \).
• For instance, when \( x = 0 \), \( f(0) = -2 \), providing the point \( (0, -2) \).
• Plot these points on a coordinate plane.
• Draw a smooth curve through the points, starting at the lowest point and extending upwards to the right.

Graphing functions is a powerful tool for understanding the essential characteristics of a function, such as slopes, inflection points, and trends. It provides insight that numerical analysis alone might miss.