Domain And Intercepts Of C(x) = (4x + 3) / (x + 4) A Comprehensive Guide

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This article delves into the analysis of the rational function C(x) = (4x + 3) / (x + 4), focusing on determining its domain and identifying its intercepts. Understanding these fundamental aspects is crucial for comprehending the behavior and characteristics of this function.

(a) Determining the Domain of C(x) = (4x + 3) / (x + 4)

The domain of a function encompasses all possible input values (x-values) for which the function produces a valid output. For rational functions, which are functions expressed as a ratio of two polynomials, a key consideration in determining the domain is identifying any values that would make the denominator equal to zero. Division by zero is undefined in mathematics, so any x-values that result in a zero denominator must be excluded from the domain. Therefore, to determine the domain of our given rational function, C(x) = (4x + 3) / (x + 4), we need to find the values of x that make the denominator (x + 4) equal to zero.

To find these values, we set the denominator equal to zero and solve for x:

x + 4 = 0

Subtracting 4 from both sides, we get:

x = -4

This indicates that when x is equal to -4, the denominator of the function becomes zero, rendering the function undefined at that point. Consequently, x = -4 must be excluded from the domain of C(x). In other words, the function is defined for all real numbers except for x = -4. The exclusion of x = -4 creates a vertical asymptote on the graph of the function, signifying a point where the function approaches infinity (or negative infinity) as x gets closer to -4. Understanding these asymptotic behaviors is pivotal in grasping the overall graph and nature of rational functions.

Therefore, we can express the domain of C(x) in several ways. Using set notation, we can write it as {x | x ∈ ℝ, x ≠ -4}, which reads as "the set of all x such that x is a real number and x is not equal to -4." Alternatively, we can use interval notation to represent the domain as (-∞, -4) ∪ (-4, ∞). This notation signifies the union of two intervals: all real numbers less than -4 and all real numbers greater than -4. The parenthesis around -4 indicates that -4 is excluded from both intervals. Correctly identifying the domain is the bedrock for further analyses such as finding intercepts, asymptotes, and ultimately graphing the rational function.

In summary, the domain of the rational function C(x) = (4x + 3) / (x + 4) is all real numbers except x = -4. This exclusion arises from the fact that the denominator, x + 4, becomes zero when x = -4, leading to an undefined function value. Recognizing and understanding such restrictions is a critical step in working with rational functions.

(b) Identifying Intercepts of C(x) = (4x + 3) / (x + 4)

Intercepts are the points where the graph of a function intersects the coordinate axes. Specifically, the x-intercepts are the points where the graph intersects the x-axis, and the y-intercept is the point where the graph intersects the y-axis. Finding the intercepts of a function provides valuable information about its behavior and helps in sketching its graph. For rational functions like C(x) = (4x + 3) / (x + 4), intercepts are relatively straightforward to compute.

Finding the x-intercept(s)

The x-intercepts are the points where the function's value, C(x), is equal to zero. In other words, we are looking for the x-values that satisfy the equation C(x) = 0. For a rational function to be zero, its numerator must be zero (while the denominator is non-zero). Therefore, to find the x-intercepts of C(x) = (4x + 3) / (x + 4), we need to solve the equation:

4x + 3 = 0

Subtracting 3 from both sides, we get:

4x = -3

Dividing both sides by 4, we find:

x = -3/4

Thus, the x-intercept occurs at x = -3/4. This means the graph of C(x) intersects the x-axis at the point (-3/4, 0). To verify this, we substitute x = -3/4 into the original function: C(-3/4) = (4(-3/4) + 3) / ((-3/4) + 4) = (-3 + 3) / (13/4) = 0 / (13/4) = 0. This confirms that -3/4 is indeed an x-intercept. The x-intercept provides insight into where the function crosses or touches the horizontal axis.

Finding the y-intercept

The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when x = 0. To find the y-intercept of C(x) = (4x + 3) / (x + 4), we substitute x = 0 into the function:

C(0) = (4(0) + 3) / (0 + 4) = 3 / 4

Therefore, the y-intercept occurs at y = 3/4. This means the graph of C(x) intersects the y-axis at the point (0, 3/4). The y-intercept is valuable as it signifies the function's value when the input is zero, offering a crucial point for graphical representation.

In conclusion, the intercepts of the rational function C(x) = (4x + 3) / (x + 4) are the x-intercept at (-3/4, 0) and the y-intercept at (0, 3/4). These intercepts, along with the domain and any asymptotes, provide a comprehensive understanding of the function's behavior and facilitate accurate graphing. Identifying the intercepts is a core procedure when analyzing various types of functions, enabling a better visual and analytical comprehension of their properties.

Summary: Domain and Intercepts of C(x) = (4x + 3) / (x + 4)

In summary, for the rational function C(x) = (4x + 3) / (x + 4), we have determined the following:

  • Domain: All real numbers except x = -4. This is because x = -4 makes the denominator zero, which is undefined. The domain can be expressed in set notation as {x | x ∈ ℝ, x ≠ -4} or in interval notation as (-∞, -4) ∪ (-4, ∞).
  • x-intercept: The x-intercept is located at the point (-3/4, 0). This was found by setting the numerator of the function (4x + 3) equal to zero and solving for x.
  • y-intercept: The y-intercept is located at the point (0, 3/4). This was found by substituting x = 0 into the function.

Understanding the domain and intercepts provides a solid foundation for further analysis, such as identifying asymptotes and sketching the graph of the function. These key features are fundamental building blocks in the comprehensive study of rational functions. The ability to efficiently find domains and intercepts is indispensable in mathematics, empowering one to deeply understand functions and solve related problems effectively. By mastering these concepts, we not only improve our understanding of rational functions but also enhance our mathematical prowess in general. This knowledge is especially valuable in fields that require modeling with functions, including physics, engineering, and economics. Therefore, a thorough grasp of the techniques discussed here is a worthwhile endeavor for anyone pursuing quantitative disciplines.