Problem 41 - College Algebra

Question

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

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty, +\infty) \) and the graph is a horizontal line at \( y=3 \) for \( x<0 \) and the square root function for \( x\geq0 \).

Step 1: Understand the Function

The function is piecewise defined, meaning it behaves differently on separate intervals of the input variable, \( x \). Specifically, \( f(x) = 3 \) when \( x < 0 \) and \( f(x) = \sqrt{x} \) when \( x \geq 0 \). This implies we have two distinct expressions describing the function based on the value of \( x \).

Step 2: Determine the Domain

The domain of the function consists of all the values \( x \) can take. For the current function, \( f(x) \) is defined for all real numbers because \( f(x) = 3 \) for all \( x < 0 \) and \( f(x) = \sqrt{x} \) for all \( x \geq 0 \). Therefore, the domain in interval notation is \(( -\infty, +\infty)\).

Step 3: Analyzing the Function on Each Interval

For \( x < 0 \), the function outputs a constant value of 3. Therefore, for negative values of \( x \), the graph will be a horizontal line at \( y = 3 \). For \( x \geq 0 \), the function's output is \( \sqrt{x} \) which is a standard square root function starting from \( x = 0 \). This portion of the graph will start at the origin and increase as \( x \) increases.

Step 4: Sketch the Graph

On the graph, plot a horizontal line at \( y = 3 \) for \( x < 0 \). The segment of the graph representing \( \sqrt{x} \) will begin at \( (0,0) \) and curve upwards as \( x \) increases, showing the square root function. Use an open circle at \((0,3)\) to indicate \( f(x) = 3 \) is not inclusive at \( x = 0 \), and a filled circle at \((0,0)\) for the square root function to indicate \( x = 0 \) is inclusive.

Key Concepts

Domain in Interval Notation

When working with functions, specifying the domain is crucial, as it tells us the set of input values over which the function is defined. In interval notation, the domain of a function is indicated using brackets and parentheses:

• Round parentheses (...) indicate that an endpoint is not included in the interval.
• Square brackets [...] indicate that an endpoint is included in the interval.

For instance, consider the piecewise function:\[f(x)=\left\{\begin{array}{ll}{3} & {\text { if } x<0} \ {\sqrt{x}} & {\text { if } x \geq 0}\end{array}\right.\]The function is defined everywhere for real numbers, since for \( x < 0 \), the function \( f(x) = 3 \) is a constant, and for \( x \geq 0 \), the function \( f(x) = \sqrt{x} \) is defined for non-negative values of \( x \). Therefore, the domain of the entire function is \[(-\infty, +\infty)\].

Graph Sketching

Creating a visual representation of a function can greatly simplify the understanding of its behavior and characteristics. To effectively sketch a graph of a piecewise function, follow these steps:

• Identify the different intervals and the corresponding expressions of the function.
• Determine if there are any endpoints on each interval, and whether these points are included or excluded.
• Sketch each part of the function within its given interval. Clearly mark where the function transitions from one part to another.
• Use open and closed circles to show whether endpoints are inclusive or exclusive.

For the provided function example, sketching entails a horizontal line at \( y = 3 \) for \( x < 0 \), marking this with an open circle at \( (0, 3) \) to show that \( x=0 \) is not included. Then, for \( x \geq 0 \), begin drawing \( \sqrt{x} \) starting at \( (0, 0) \) with a closed circle, curving upwards to represent increasing values.

Square Root Function

The square root function, typically expressed as \( f(x) = \sqrt{x} \), features unique properties:

• Its domain includes only non-negative numbers, \( x \geq 0 \), as square roots of negative numbers are not real.
• The graph of \( \sqrt{x} \) begins at the origin \((0,0)\), and moves upwards in a smooth curve.
• As \( x \) increases, the growth of \( \sqrt{x} \) slows down due to the nature of square roots.

In the piecewise function example, \( f(x) = \sqrt{x} \) starts at \( x=0 \) and is included, creating continuity from zero as it progresses.

Continuous Function

A continuous function is one that has no breaks, jumps, or holes in its graph. Continuity means you can draw the function without lifting your pencil from the paper. In terms of piecewise functions, continuity is assessed at points where different pieces of the function meet.

For our function:

• Naturally, \( f(x) = 3 \) for \( x < 0 \) is continuous within its interval because it is a constant line.
• \( f(x) = \sqrt{x} \) is likewise continuous within the domain \( x \geq 0 \).
• The function needs examination at \( x = 0 \), the transition point between the two expressions.

Here, continuity is not achieved at \( x = 0 \) since the value jumps from 3 to 0. This is marked on the graph using open and closed circles, highlighting the lack of continuity. This is a classic representation of how piecewise functions can handle continuity, sometimes producing discontinuous graphs where different rules meet.