Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. \(f(x)=\left\\{\begin{array}{cl}3 & \text { if } x<0 \\ \sqrt{x} & \text { if } x \geq 0\end{array}\right.\)

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); graph: horizontal line at \( y = 3 \) for \( x < 0 \), square root curve for \( x \geq 0 \).

Step 1: Understand the Function

The function given is a piecewise function with two parts. The first part, \( f(x) = 3 \), is valid for \( x < 0 \). The second part, \( f(x) = \sqrt{x} \), is used when \( x \geq 0 \). Each part of the function corresponds to a different condition for the input \( x \).

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (\( x \)) for which the function is defined. In this piecewise function, for \( x < 0 \), \( f(x) \) is defined and equals 3, and for \( x \geq 0 \), \( f(x) \) is defined as \( \sqrt{x} \). Thus, the domain of \( f(x) \) includes all real numbers, which can be expressed in interval notation as \((-\infty, \infty)\).

Step 3: Sketch the Graph for Part 1

For \( x < 0 \), the function \( f(x) = 3 \) is constant. On a graph, this is represented as a horizontal line at \( y = 3 \) that extends to the left (for negative values), stopping just before \( x = 0 \). Ensure your line has an open circle at \( x = 0 \) to denote that this part of the function does not include \( x = 0 \).

Step 4: Sketch the Graph for Part 2

For \( x \geq 0 \), the function is \( f(x) = \sqrt{x} \). This part of the graph starts at \( (0, 0) \) and extends to the right. Note that the point at \( x = 0 \) is filled in, as \( x = 0 \) is included in this part of the function. The graph will show a typical square root curve, starting at the origin and rising slowly to the right.

Step 5: Combine the Graphs

To complete the graph, combine the two parts. The left part is the horizontal line at \( y = 3 \) for \( x < 0 \), and the right part is the square root curve starting from the origin. It is important to clearly show the open circle at \( x = 0 \) on the line \( y = 3 \) and the filled circle starting the square root curve at the origin.

Key Concepts

Domain

The domain of a function encapsulates all permissible input values. For the piecewise function given, which is defined as:

• 3, when \( x < 0 \)
• \( \sqrt{x} \), when \( x \geq 0 \)

each part has specific conditions. Therefore, the overall domain encompasses all real numbers since:

• For \( x < 0 \), the piece \( f(x) = 3 \) is valid.
• For \( x \geq 0 \), \( f(x) = \sqrt{x} \) is valid and continuous at \( x = 0 \).

The domain in interval notation is expressed as \(( -\infty, \infty )\). This indicates the function accepts every real number, from negative infinity to positive infinity, without interruption. Understanding domains is crucial as it provides insight into where the function can be evaluated.

Interval Notation

Interval notation provides a concise way to describe the set of values within a given interval. It consists of brackets and parentheses to indicate inclusion or exclusion of endpoints.

• Square brackets \([\ ]\) signify that an endpoint is included in the interval.
• Parentheses \((\ )\) show that an endpoint is excluded.

For the piecewise function \( f(x) \), its domain in interval notation is \(( -\infty, \infty )\), meaning there are no restrictions on \( x \) from negative infinity to positive infinity. This form succinctly conveys the concept that \( x \) can be any real number, a fundamental aspect when defining domains in mathematical contexts.

Graph Sketching

Graph sketching is an essential tool in visualizing mathematical functions. It provides a visual representation that aids in understanding how a function behaves.To sketch the graph of a piecewise function like \( f(x) \), follow these steps:- **Identify each piece of the function.** Here, it's a constant value for \( x < 0 \) and a square root function for \( x \geq 0 \).- **Draw each section accurately.** For \( x < 0 \), draw a horizontal line at \( y = 3 \) with an open circle at \( x = 0 \), illustrating that this segment does not include zero.- For \( x \geq 0 \), begin the square root curve at the origin and move right, ensuring the point at \((0, 0)\) is filled in, reflecting its inclusion.- **Combine each section** clearly on a shared coordinate plane. Make sure the transition between different expressions of the piecewise function is shown with precise endpoints.Graph sketching involves merging analytical solutions with visual elements, enhancing comprehension and communication of mathematical ideas.

Square Root Function

The square root function, denoted as \( \sqrt{x} \), is a widely used function in mathematics, noteworthy for its distinct curve.- **Definition and Basic Form:** The square root of a number \( x \) is a value that, when multiplied by itself, yields \( x \). It is typically written as \( \sqrt{x} \).- **Domain and Range:** The domain of \( \sqrt{x} \) includes non-negative real numbers (\( x \geq 0 \)). The range is also non-negative, extending from zero to positive infinity.For the piece \( x \geq 0 \) in our function, \( f(x) = \sqrt{x} \) begins at the origin \((0,0)\). The graph shows a classic curve starting from the origin that slowly ascends to the right as \( x \) increases. Understanding this helps in recognizing common functions and their properties, essential for further mathematical studies.