Problem 37 - Algebra and Trigonometry

Question

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{x} & {\text { if } x \leq 0} \\ {x+1} & {\text { if } x>0}\end{array}\right.\)

Step-by-Step Solution

Answer

Graph the linear pieces: one with \( f(x)=x \) for \( x \leq 0 \), and another with \( f(x)=x+1 \) for \( x > 0 \).

Step 1: Identify the Piecewise Function Parts

The function is defined in two parts: \( f(x) = x \) for \( x \leq 0 \) and \( f(x) = x + 1 \) for \( x > 0 \). This means we'll be dealing with two separate linear equations, depending on the domain of \( x \).

Step 2: Graph the First Part

For \( f(x) = x \) where \( x \leq 0 \), this is a linear function with a slope of 1 passing through the origin. Graph this line in the negative \( x \)-axis and a closed circle at (0, 0) to indicate that this part of the function includes zero.

Step 3: Graph the Second Part

For \( f(x) = x + 1 \) where \( x > 0 \), this function is also linear with a slope of 1 but starts at y-intercept (0, 1). Plot this line starting right after \((0,0)\) with an open circle to indicate that zero is not included in this piece.

Step 4: Combine the Graphs

Combine the two graphs by ensuring the transition from \( x \leq 0 \) to \( x > 0 \) is smooth and accurate. Check that the graph correctly depicts both parts of the piecewise function with appropriate open and closed circles at x=0.

Key Concepts

Linear Functions

Linear functions are fundamental in mathematics and describe relationships with a constant rate of change. In simple terms, they are functions that graph as a straight line. The general form of a linear function is \( y = mx + b \), where:

• \( m \) is the slope of the line, representing how steep or flat the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

When dealing with piecewise functions like the one in the original exercise, each piece is usually a linear function. For example:

• \( f(x) = x \) indicates a slope of 1 passing through the origin.
• \( f(x) = x + 1 \) also has a slope of 1, but it is shifted up, starting from the y-intercept at 1.

Understanding how to plot these lines and where they intersect the axes helps in visualizing the complete piecewise function.

Graphing Functions

Graphing a function is a powerful technique in mathematics that visually represents the relationship described by the function. For linear functions, you start by identifying the slope and the y-intercept from the equation \( y = mx + b \). When graphing piecewise functions like in the provided example, it's important to:

• Graph each segment of the function individually according to its specific condition.
• Pay attention to where the graph is continuous and where it changes direction or form.
• Use open or closed circles to indicate whether endpoints are included or excluded.

Specifically for our piecewise function, you would:

• Graph the line \( y = x \) for \( x \leq 0 \) with a closed circle at \((0,0)\) because the endpoint is included.
• Graph the line \( y = x + 1 \) for \( x > 0 \) with an open circle at \((0,1)\) because the endpoint is not included.

The goal is to ensure that the transition between pieces is represented accurately and continuously.

Domain and Range

The domain and range are key concepts in understanding a function's behavior:

• The **domain** refers to all possible input values (\( x \)-values) for which the function is defined.
• The **range** refers to all possible output values (\( y \)-values) that the function can produce.

In our example of a piecewise function:

• The domain is all real numbers because either piece of the function is defined for every real number.
• The range varies depending on the function parts. For \( f(x) = x \), the range is all real numbers less than or equal to 0. For \( f(x) = x + 1 \), the range is all real numbers greater than 1.

Knowing the domain and range helps in accurately graphing the function and understanding its limitations and capabilities.