Analyzing Rational Function: Discontinuities, Intercepts, Graph
Let's dive into the world of rational functions by thoroughly examining the function f(x) = (x^2 + x - 12) / (x + 4). We'll pinpoint removable discontinuities, sketch its graph, locate x-intercepts, and determine the y-intercept. This comprehensive analysis will give us a solid understanding of how this rational function behaves.
a. Determining Removable Discontinuities
When we talk about removable discontinuities, we're essentially looking for points where the function is undefined, but the discontinuity can be 'removed' by simplifying the function. These occur when a factor in the numerator and denominator cancels out.
Our function is:
f(x) = (x^2 + x - 12) / (x + 4)
First, let's factor the numerator:
x^2 + x - 12 = (x + 4)(x - 3)
So, we can rewrite the function as:
f(x) = ((x + 4)(x - 3)) / (x + 4)
Notice that the (x + 4) term appears in both the numerator and the denominator. This means we have a removable discontinuity at x = -4. To find the coordinates of this discontinuity, we need to determine the value that the function would have at x = -4 if we could 'remove' the discontinuity.
After canceling the (x + 4) terms, we're left with:
f(x) = x - 3, x ≠-4
Now, we can plug x = -4 into the simplified function to find the y-coordinate of the removable discontinuity:
f(-4) = -4 - 3 = -7
Therefore, the coordinates of the removable discontinuity are (-4, -7). This means that the function behaves like the line y = x - 3, except there's a 'hole' at the point (-4, -7). Visualizing this helps in understanding the function's graph and behavior around this point. Understanding removable discontinuities is crucial for further analysis in calculus, especially when dealing with limits and derivatives. These discontinuities often arise in real-world applications, such as modeling physical systems where certain conditions might lead to undefined states but can be mathematically 'smoothed' over for analysis purposes. Identifying and addressing these discontinuities allows for a more complete and accurate understanding of the function's overall behavior and its applicability to various problems. This detailed process underscores the importance of factorization and simplification in analyzing rational functions, ensuring a clear and precise representation of their properties and characteristics.
b. Sketching the Graph
To sketch the graph of f(x) = (x^2 + x - 12) / (x + 4), we'll use the information we've already gathered and a few additional considerations.
- Simplified Function: We know that f(x) = x - 3, except at x = -4.
- Removable Discontinuity: We have a hole at (-4, -7).
- y-intercept: We'll find this in part (d), but it's useful for sketching.
- x-intercept: We'll find this in part (c), also useful for sketching.
Since the function simplifies to a linear equation (y = x - 3), the graph will be a straight line with a slope of 1 and a y-intercept of -3 (which we'll confirm later). However, we must remember the hole at (-4, -7). To sketch the graph:
- Draw the line y = x - 3.
- At the point x = -4, draw a small open circle to indicate the removable discontinuity.
Important Considerations for Sketching:
- Asymptotes: Since the discontinuity is removable and we've simplified the function, there are no vertical asymptotes. The original denominator, (x + 4), becomes zero at x = -4, but the factor cancels out, so it doesn't create a vertical asymptote.
- End Behavior: As x approaches positive or negative infinity, the function behaves like the line y = x - 3. There are no horizontal or slant asymptotes because the degree of the numerator and denominator were initially the same, but the function simplifies to a linear form.
By plotting a few points on the line y = x - 3 and marking the hole at (-4, -7), you can accurately sketch the graph of the function. The graph provides a visual representation of the function's behavior, clearly showing the linear relationship with a single point of discontinuity. Accurately sketching the graph of a function like f(x) involves more than just plotting points; it requires understanding the underlying principles of function behavior, transformations, and the implications of discontinuities. Being mindful of asymptotes, intercepts, and the end behavior ensures that the sketch accurately represents the function's characteristics, providing valuable insights for both mathematical analysis and practical applications. Sketching a graph is not merely a visual aid but a tool for problem-solving and conceptual understanding.
c. Finding x-intercepts
To find the x-intercepts, we need to find the values of x for which f(x) = 0. In other words, we need to solve the equation:
(x^2 + x - 12) / (x + 4) = 0
A rational function is equal to zero only when its numerator is equal to zero (and the denominator is not zero at the same point). So, we need to solve:
x^2 + x - 12 = 0
We already factored this in part (a):
(x + 4)(x - 3) = 0
This gives us two possible solutions:
- x + 4 = 0 => x = -4
- x - 3 = 0 => x = 3
However, we must remember that x = -4 is a removable discontinuity. The function is not defined at x = -4, so it cannot be an x-intercept. Therefore, the only x-intercept is x = 3. So the x-intercept is (3, 0).
Understanding x-intercepts is vital in various fields, including physics, engineering, and economics, where they often represent critical points such as equilibrium states or break-even points. By accurately finding and interpreting x-intercepts, analysts and practitioners can gain valuable insights into the behavior and implications of mathematical models in real-world scenarios. This highlights the practical relevance of mastering the techniques for finding x-intercepts and applying them to solve complex problems across different domains. Therefore, being thorough and precise is essential when determining x-intercepts, as they serve as fundamental reference points in understanding the function's characteristics and its applications in diverse contexts. Moreover, the process of finding x-intercepts reinforces essential algebraic skills, such as factoring, solving equations, and considering domain restrictions, which are valuable in mathematical analysis and problem-solving.
d. Finding the y-intercept
To find the y-intercept, we need to find the value of f(0). In other words, we need to evaluate the function when x = 0:
f(0) = (0^2 + 0 - 12) / (0 + 4)
f(0) = -12 / 4
f(0) = -3
So, the y-intercept is (0, -3).
The y-intercept represents the point where the function's graph intersects the y-axis, offering a crucial reference point for understanding its behavior. In practical terms, the y-intercept often represents the initial value or starting point of a system modeled by the function. For instance, in a cost function, the y-intercept might represent the fixed costs incurred before any units are produced. Similarly, in a population growth model, the y-intercept indicates the initial population size. Finding the y-intercept is often straightforward, involving substituting x = 0 into the function and evaluating the result. This process provides a quick and direct way to determine the function's value at this key point. However, it's essential to ensure that the function is defined at x = 0, as some functions may have restrictions or discontinuities at this point. Understanding the significance and implications of the y-intercept enhances our ability to interpret and apply mathematical models to real-world scenarios, making it a valuable tool in various disciplines. Therefore, taking the time to accurately determine and interpret the y-intercept is crucial for gaining a comprehensive understanding of the function's properties and its relevance in different contexts. Furthermore, the y-intercept serves as a fundamental anchor point when graphing the function, aiding in visualizing its behavior and relationship to other points and features. Analyzing the function, especially when combined with other key features such as x-intercepts and asymptotes, provides a holistic view of its characteristics and its potential applications.
In summary, by factoring, simplifying, and carefully analyzing the given rational function, we've identified its removable discontinuity, sketched its graph, and found its x and y-intercepts. This comprehensive approach showcases the power of algebraic manipulation and graphical representation in understanding the behavior of rational functions.
For further reading, you might find valuable insights on rational functions at Khan Academy's page on rational functions.
