Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is a horizontal line at \(y=1\) for \(x\leq 1\), and a line with a slope 1 starting from \(x=1\), which is open at \((1,2)\).

Step 1: Step 1: Identify Sections

The given piecewise function is defined by two different rules based on the value of \(x\). The function consists of two parts: \(f(x) = 1\) if \(x \leq 1\) and \(f(x) = x + 1\) if \(x > 1\). We need to analyze and sketch each of these separately.

Step 2: Sketch First Section

For the first section of the piecewise function, \(f(x) = 1\) when \(x \leq 1\). This represents a horizontal line at \(f(x) = 1\) that continues to the left from \(x = 1\). Since the point at \(x=1\) is included in this part of the function, indicate it with a solid dot at \((1,1)\).

Step 3: Sketch Second Section

For the second section, \(f(x) = x + 1\) when \(x > 1\), which represents a line starting just to the right of \(x = 1\). The y-intercept of this line is \(1\) (since when \(x = 0\), which does not apply here since \(x > 1\), \(y = 1\)). However, its relevant starting value at \(x = 1\) is \(f(1) = 2\). Draw this line with an open circle at \((1,2)\) to denote that it does not include this point and extend the line upwards to the right.

Step 4: Draw and Label the Graph

Combine the sections from Step 2 and Step 3 onto a single set of axes. Clearly mark the horizontal segment at \(y=1\) extending to the left from \(x=1\), and the diagonal line starting just right of \(x=1\) with an open circle on \((1,2)\) and continuing upwards. Label key points and curves to maintain clarity.

Key Concepts

Graphing Functions

Graphing functions involves plotting points that satisfy the function's equation on a coordinate plane. This helps visualize the behavior of the function across different values of the input variable, often denoted as \(x\). By understanding how to graph, students gain insights into the function's properties, such as slope, intercepts, and continuity.
When starting to graph any function, it's crucial to:

• Identify the type of function and its general shape
• Determine key points such as intercepts and turning points
• Identify any restrictions, such as domains or piecewise sections

In our exercise, the function is piecewise, meaning it has different expressions based on the input. Graphing such functions requires careful attention to the transitions between these sections.

Piecewise-Defined Functions

Piecewise-defined functions are special kinds of functions that have multiple sub-functions, each applied to different parts of the function's domain. This means that the rules governing the output of the function can change depending on the value of the input.
For instance, in our function:

• When \(x \leq 1\), the output is always \(1\). This creates a constant horizontal line across the graph.
• When \(x > 1\), the function follows the rule \(f(x) = x + 1\), resulting in a line with a slope of 1 and a y-intercept that would be 1 if extended.

Understanding these sections is critical to properly graphing the function. It requires recognizing the transition point (here, \(x = 1\)) and depicting any inclusions or exclusions using solid or open dots to indicate if points lie within the function's range.

Function Sketching

Function sketching is the art of drawing a rough graph of a function without calculating all points meticulously. It's a skill that combines mathematical understanding with creative estimation.

• Begin by identifying the major sections of the function, like in our example, where we have defined parts for different values of \(x\).
• Mark the transition points where these sections change, emphasizing whether points at these transitions are inclusive (solid dot) or exclusive (open dot).
• Accentuate the shape: horizontal lines for constants and sloped lines for linear growth or decay.

By following these steps, you capture the essence of the function's behavior visually, providing insights into not only how the function looks but also how it behaves as \(x\) changes across its domain.