Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function on the interval \([-4,4]\). Does the graph of the function appear to be continuous on this interval? Is the function in fact continuous on \([-4,4] ?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. \(f(x)=\frac{x^{2}+x}{x}\)

Step-by-Step Solution

Answer

Upon graphing the simplified function \(f(x)= x + 1\) on a graphing utility, it appears visually continuous on the interval \([-4,4]\). However, upon closer analytical inquiry, despite the function \(f(x)=\frac{x^{2}+x}{x}\) simplifying to \(f(x)= x + 1\), this simplification is undefined at \(x=0\), thus making the function discontinuous at \(x=0\) and hence not continuous over the full interval \([-4,4]\). This exercise highlights the importance of examining a function analytically as well as graphically.

Step 1: Simplify the Function

The given function \(f(x)=\frac{x^{2}+x}{x}\) simplifies to \(f(x)= x + 1\), for \(x \neq 0\). Therefore, we only need to graph the function \(f(x)= x + 1\).

Step 2: Determine the Graph

By graphing \(f(x)= x + 1\) using a graphing utility, you can see that the function is a straight line with a slope of 1 and y-intercept of 1. It extends continuously from -4 to 4.

Step 3: Analyze the Continuity of the Graph

Looking at the graph, it appears to be continuous on the interval \([-4,4]\). However, this does not confirm that the function itself is continuous on that interval since the original function is undefined at \(x=0\).

Step 4: Confirm the Continuity Analytically

To analytically confirm the continuity of the function on the interval \([-4,4]\), we observe that although the function \(f(x)=\frac{x^{2}+x}{x}\) simplifies to \(f(x)= x + 1\), the simplification is not defined at \(x=0\). Therefore, the function is not continuous at \(x=0\) and hence not continuous on \([-4,4]\).

Step 5: Explain the Importance of Analysing the Function Both Graphically and Analytically

Examining a function both graphically and analytically is very important. A graphical analysis provides immediate visual understanding of the function's behavior over the defined interval. Analytical analysis, on the other hand, confirms the findings from the graphical inspection and may reveal aspects of the function's nature that are not immediately apparent from the graph, such as discontinuities at specific points.

Key Concepts

Continuity of Functions

The continuity of a function is a crucial concept in calculus and mathematics in general. A function is considered continuous over an interval if it can be drawn without lifting the pen off the paper on a graph. In simpler terms, for a function to be continuous at a point, it must satisfy three conditions: the function is defined at the point, the limit of the function exists as the point is approached from both sides, and the limit equals the function value at that point.
For the given function, \[ f(x) = \frac{x^2 + x}{x} \], the simplification to \[ f(x) = x + 1 \] is defined everywhere except at \( x = 0 \) due to division by zero. This break in continuity at \( x = 0 \) means the function is not continuous over the entire interval \([-4, 4]\). It's important to explore both the simplified version and its conditions for validity to assess continuity properly.

Graphical Analysis

Graphical analysis involves using a graph to explore the behavior of a function visually. It provides a quick overview of how the function behaves across its domain.
When graphing \( f(x) = x + 1 \), we see a straight line with a slope of 1. The line stretches smoothly from \( x = -4 \) to \( x = 4 \), with a \( y \)-intercept at \( y = 1 \).
Graphs are great for spotting general trends, slopes, intercepts, and overall shape of a function. However, they might not always show small details like discontinuities at specific points. Therefore, complementing graphical analysis with other types of analysis is key for a thorough understanding.

Analytical Analysis

Analytical analysis dives deeper into the mathematical functions and their properties by evaluating their algebraic expressions. It allows us to confirm what we observe graphically and discover hidden traits.
For \[ f(x) = \frac{x^2 + x}{x} \], simplifying to \( f(x) = x + 1 \) analytically shows that all operations and outcomes alter near \( x = 0 \). By analyzing algebraic expressions, you evaluate limits, identify undefined areas, and verify or refute continuity across given intervals. Such analytical insights reveal that \( x = 0 \) is not just a point on a graph—it’s the point where the function is undefined, proving non-continuity at that single point.

Undefined Points

Undefined points occur in functions where certain operations, like division by zero, cannot be performed. These are points where the function does not exist or is not defined. In the provided function \[ f(x) = \frac{x^2 + x}{x} \], the division by zero at \( x = 0 \) results in such an undefined point.
Understanding undefined points is vital because they impact continuity, differentiability, and the overall behavior of the function. Analytically identifying these points through simplification and logical analysis (by checking the conditions when it becomes undefined) helps plot a more accurate character of the function over its entire range. Recognizing such points in graphical analysis might require zooming or detailed inspection, as they may not be evident from a simple look at the graph.