Question

Sketch, as best you can, the graph of a function \(f\) that satisfies all the following conditions. (a) Its domain is the interval \([0,4]\). (b) \(f(0)=f(1)=f(2)=f(3)=f(4)=1\) (c) \(\lim _{x \rightarrow 1} f(x)=2\) (d) \(\lim _{x \rightarrow 2} f(x)=1\) (e) \(\lim _{r \rightarrow 3^{-}} f(x)=2\) (f) \(\lim _{\rightarrow+} f(x)=1\)

Short answer

Draw a flat line at \(y = 1\), with small gaps at \(x = 1\) and \(x=3\) for limits.

Step-by-step solution

Understanding Domain and Points

The domain of the function is \([0,4]\), meaning the graph should only exist in this interval. The function assigns specific values to points: \(f(0)=1\), \(f(1)=1\), \(f(2)=1\), \(f(3)=1\), and \(f(4)=1\). These represent horizontal coordinates where the function's value is known.

Analyze Limit Around x=1

The function has a limit at \(x=1\) where \(\lim_{x\to 1}f(x)=2\). This tells us to expect the graph to approach the value 2 as \(x\) approaches 1 from either side; however, the function value is 1 at \(x=1\). This indicates a removable discontinuity or a jump at \(x=1\) where you can represent this with a small gap or circle in the graph at (1,2).

Study Limit Around x=2

\(\lim_{x\to 2}f(x)=1\) implies that as \(x\) approaches 2, the function approaches 1. Here, no discontinuity is indicated as both the limit and actual function value agree. The graph touches the point at (2,1) smoothly without an apparent break.

Investigate Limit Around x=3-

Here \(\lim_{x\to 3^{-}}f(x)=2\) indicates that coming from the left side the function value approaches 2. However, the function equals 1 at \(x=3\). Hence, there might be a discontinuous drop from the left of 3 before the function reaches \(f(3)=1\). Represent the left side of (3,2) with a circle before the graph jumps back down.

Limit Behavior for x>3

\(\lim_{x\to 3^{+}}f(x)=1\), strengthens the observation from the previous step—Post 3, the function approaches and continues from the value 1 smoothly without any jumps or discontinuities. The graph after 3 continues straight from the point (3,1) to (4,1).

Final Graph Rendering

Combine all these information into a sketch: a horizontal baseline at \(y=1\) holds for \(x=0,2,3,4\) with small disruptions around \(x=1\) and \(x=3\). At \(x=1\), place an open circle at (1,2) representing the limit and a solid dot below at (1,1). Similarly, place a little overlap to denote \(3,2\) by an open circle and continue the existing line to (3,1) with a filled rectangle.

Key concepts

Function Limits

Understanding function limits is essential for sketching and interpreting graphs. A limit describes the value that a function approaches as the input approaches a specific point. For example, in the exercise, at various points like x=1, x=2, and x=3, the graph approaches certain y-values.

• If \( \lim_{x \to 1} f(x) = 2 \), the function gets closer to a y-value of 2 as x approaches 1.
• Even if the function value at that specific point is different (such as \( f(1) = 1 \)), the limit tells us how the function behaves when nearing that point.

Remembering that limits help us understand behavior near a point but don’t have to match the actual function value at that point is key.

Discontinuities

A discontinuity in a function is where the graph has breaks, jumps, or holes. These are essential to identify when sketching graphs.

• In a continuous function, you'd expect a smooth curve with no interruptions.
• However, discontinuities like at \( x = 1 \) and \( x = 3 \) in this exercise signal places where the graph might deviate from the usual path.

Recognizing these helps in visualizing the graph correctly and understanding any sudden changes in a function's behavior.

Removable Discontinuities

Removable discontinuities happen when a function isn’t defined at a point, but limit values could allow for a smooth patch if the point were redefined.

• For \( x = 1 \), the limit indicates \( \lim_{x \to 1} f(x) = 2 \), a separate value from the defined \( f(1) = 1 \).
• This removable discontinuity is represented by an open circle at (1,2) on the graph, indicating the gap.

The discontinuity is 'removable' because adjusting the function value at this point could fill the gap and make it continuous.

Limit Behavior

Examining limit behavior can tell us a lot about how a function acts around critical points.

• Understanding one-sided limits helps predict jumps or smooth transitions.
• For \( x = 3 \), the left limit is \( \lim_{x \to 3^{-}} f(x) = 2 \), showing a different approach value than when \( 3 \) is exactly reached.

Limit behavior helps in accurately drawing graph components, predicting slopes, or points of change before and after given x-values.