Problem 49 - Algebra A Combined Function

Question

Graph: \(f(x)=\left\\{\begin{aligned}-\frac{1}{2} x & \text { if } x \leq 0 \\\ x+1 & \text { if } 02 \end{aligned}\right.\)

Step-by-Step Solution

Answer

Plot three line segments, transitioning at 0 and 2.

Step 1: Understanding the Piecewise Function

The function \( f(x) \) is piecewise, meaning it's defined by different expressions based on the value of \( x \). It has three parts: \( f(x) = -\frac{1}{2}x \) for \( x \leq 0 \), \( f(x) = x + 1 \) for \( 0 < x \leq 2 \), and \( f(x) = 2x - 1 \) for \( x > 2 \).

Step 2: Identify Key Points and Transitions

Determine key points: at \( x = 0 \), the function transitions from \( -\frac{1}{2}x \) to \( x + 1 \), and at \( x = 2 \), it transitions from \( x + 1 \) to \( 2x - 1 \). Calculate function values at these key points.

Step 3: Calculate Function Values at Boundary Points

For \( x = 0 \), \( f(0) = -\frac{1}{2}(0) = 0 \). For \( x = 2 \), \( f(2) = 2 + 1 = 3 \). Check for any discontinuities by verifying left and right limits at these transition points.

Step 4: Graph Each Piece

Draw each piece separately. For \( x \leq 0 \), plot \( f(x) = -\frac{1}{2}x \), which is a straight line passing through origin and extending into negative \( x \). For \( 0 < x \leq 2 \), plot \( f(x) = x + 1 \), a line with a slope of 1 starting just above \( x = 0 \). For \( x > 2 \), graph \( f(x) = 2x - 1 \), a line starting at \( x = 2 \) and continuing.

Step 5: Combine Pieces into Single Graph

Combine all the lines into a single graph, marking the points of transition. Ensure the entire graph accurately reflects changes at \( x = 0 \) and \( x = 2 \). Label the graph clearly.

Key Concepts

Function Graphing

Piecewise functions are represented by separate graphs, each corresponding to a specific segment of the function definition. In the given function, we have three distinct parts. Therefore, when graphing, visualize each segment separately first.

• For the segment where \( x \leq 0 \), graph the line \( f(x) = -\frac{1}{2}x \). This part of the function extends from \( x = -\infty \) to \( x = 0 \).
• Next, for \( 0 < x \leq 2 \), plot the line \( f(x) = x + 1 \). This part begins just after \( x = 0 \) and continues up to \( x = 2 \).
• Finally, for \( x > 2 \), graph \( f(x) = 2x - 1 \). This section picks up from \( x = 2 \) and extends beyond.

Separate these segments clearly and then merge them to form the entire graph. Always label pieces distinctly and join them by marking boundary points.

Discontinuities

Discontinuities in piecewise functions occur where the function does not seamlessly transition from one segment to another. When graphing, these are found at transition points—where one piece stops and another begins.
For our function:

• Check at \( x = 0 \): As you transition from \( f(x) = -\frac{1}{2}x \) to \( f(x) = x + 1 \), compute \( f(0) \) for both sides. Both values hold at \( 0 \), resulting in no discontinuity at this point.
• At \( x = 2 \): Compare values from the pieces \( f(x) = x + 1 \) and \( f(x) = 2x - 1 \). Again, both limit values at \( x = 2 \) match, confirming a continuous point—no jump occurs.

Identifying discontinuities helps ensure the graph accurately reflects the function's behavior across its domain.

Boundary Points

Boundary points in piecewise functions are the values of \( x \) where one function segment stops and another begins. These points are crucial for understanding function behavior and ensuring correct graphing.

• In our function, the boundary points are \( x = 0 \) and \( x = 2 \). At these points, evaluate the function to see what value each segment holds and how they connect.
• For \( x = 0 \): The value of the left segment is \( f(0) = 0 \), and the right starts its ascent immediately after, creating a seamless transition.
• For \( x = 2 \): At this junction, both segments meet at \( f(2) = 3 \), confirming a smooth graph across this point.

Mark these boundary points on the graph to aid in drawing smooth transitions between segments.

Graph Transitions

Graph transitions occur at boundary points where the behavior of a piecewise function might change. When plotting, note these shifts keenly to capture the function's true shape.

• When transitioning at \( x = 0 \) from \( f(x) = -\frac{1}{2}x \) to \( f(x) = x + 1 \), there's smoothness as both pieces join seamlessly at the same value.
• Moving to \( x = 2 \), the graph transitions from \( f(x) = x + 1 \) to \( f(x) = 2x - 1 \) without a jump, as distinctions between the pieces have been calculated to join without gaps.

Accurate plotting ensures the function's continuity and accurately maps each transition point, avoiding misalignments or inaccuracies that could distort understanding.