Problem 5 - Calculus

Question

In Exercises \(4-9,\) draw a possible graph of \(f(x) .\) Assume \(f(x)\) is defined and continuous for all real \(x\). $$\lim _{x \rightarrow \infty} f(x)=-\infty \quad \text { and } \quad \lim _{x \rightarrow-\infty} f(x)=+\infty$$

Step-by-Step Solution

Answer

Draw a continuous decreasing curve going up left and down right across the graph.

Step 1: Understanding the Limits

The problem provides two key pieces of information: 1. As \( x \) approaches \( +\infty \), \( f(x) \) approaches \( -\infty \). This means the graph falls downwards as we move towards the right. 2. As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \). This indicates the graph rises upwards as we move towards the left.

Step 2: Sketch the Basic Graph Shape

Given the limits provided, start sketching by drawing a line that begins upwards to the left (as \( x \rightarrow -\infty \)) and moves downwards toward the right (as \( x \rightarrow \infty \)). This generally resembles a decreasing curve of a function like \( f(x) = -x \).

Step 3: Consider Continuity

As the function \( f(x) \) is specified to be continuous for all real \( x \), ensure that the graph does not have any breaks or jumps. Smoothly connect the upward trend from the left to the downward trend on the right without any discontinuities.

Step 4: Finalize the Graph

Refine the sketch ensuring that as \( x \) moves towards \( -\infty \), \( f(x) \) is increasingly going upwards, and as \( x \) moves towards \( +\infty \), \( f(x) \) keeps going downwards. Make sure the overall shape reflects these behaviors and remember the curve needs to look continuous and without any breaks.

Key Concepts

Graph Sketching

Graph sketching involves drawing a curve based on the behavior of a function as described by its limits. When you hear about limits, it's all about what happens to the function as the input gets really large or really small. For the exercise you have, the limits indicate the end behaviors of the graph:

• As the input, or x-value, heads towards positive infinity, the function heads towards negative infinity. This means the graph falls downward as it moves to the right.
• Conversely, as x moves towards negative infinity, the function reaches towards positive infinity, making the graph rise as it moves to the left.

When sketching this, picture a line that enters from above on the left side and smoothly sweeps down, exiting below on the right side. This type of graph is often akin to a function with a decreasing slope, such as a line represented by f(x) = -x, but remember this is a visual approximation.

Continuity

Continuity in a function is crucial when drawing graphs as it ensures a graph can be drawn without lifting the pencil from the paper. Simply put, a continuous function has no breaks, jumps, or holes. In mathematical terms, at any point on the function, the limit must equal the function's value.
When you are drawing a graph and you know the function is continuous, you must ensure the curve doesn't have any interruptions. It should smoothly connect the rising and falling sections of the graph.

• This means every value of x has a corresponding value of f(x) without gaps.
• No sharp turns or abrupt changes should be found in the function's behavior.

For your exercise, this continuous nature means that although the graph continually rises and falls following the limits, each part must smoothly transition into the next without any discontinuities.

End Behavior

End behavior describes the behavior of the graph of a function as the input, or x-value, approaches either positive or negative infinity. This gives insight into how the function behaves far away from the center or "origin" of the graph.
In this activity, you’re focused on understanding the end behavior by analyzing limits:

• The given limit \(\lim_{x \rightarrow \infty} f(x) = -\infty\) means the graph will fall towards negative infinity as it moves to the right on the x-axis.
• The limit \(\lim_{x \rightarrow -\infty} f(x) = +\infty\) implies that the graph climbs towards positive infinity as it extends to the left.

Recognizing end behavior patterns can help you predict and draw the tail ends of the graph accurately. Remember, in a real-world context, understanding end behavior is key for making projections and understanding long-term tendencies of functions, such as economic growth trends or physical phenomena. It helps in giving a complete picture of what the function is doing at its extremes.