Rational Function Analysis Finding Domain Intercepts And Asymptotes Of F(x) = (x+4)/(4x^2+4x-3)

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In the fascinating world of mathematics, rational functions hold a special place. These functions, expressed as the ratio of two polynomials, exhibit unique behaviors and characteristics that are crucial to understand. In this article, we delve into a specific rational function, f(x) = (x+4)/(4x^2+4x-3), presented in both standard and factored forms, to unravel its domain, intercepts, and asymptotes. Our journey will involve algebraic manipulations, insightful analysis, and a touch of graphical interpretation to fully grasp the essence of this function.

Understanding the Domain of f(x)

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the primary concern is the denominator. A rational function is undefined when the denominator equals zero, as division by zero is mathematically forbidden. Therefore, to determine the domain of our function, f(x) = (x+4)/(4x^2+4x-3), we must identify the values of x that make the denominator, 4x^2 + 4x - 3, equal to zero.

The factored form of the function, f(x) = (x+4)/((2x+3)(2x-1)), provides a clear pathway to finding these excluded values. The denominator, (2x+3)(2x-1), equals zero when either 2x + 3 = 0 or 2x - 1 = 0. Solving these equations, we find that x = -3/2 and x = 1/2 are the values that make the denominator zero. These values must be excluded from the domain.

Therefore, the domain of f(x) consists of all real numbers except x = -3/2 and x = 1/2. In interval notation, we express this domain as:

(-∞, -3/2) ∪ (-3/2, 1/2) ∪ (1/2, ∞)

This notation signifies that the domain includes all numbers from negative infinity up to -3/2, then all numbers between -3/2 and 1/2, and finally all numbers from 1/2 to positive infinity. The union symbol (∪) indicates that these intervals are combined to form the complete domain.

Unveiling the Y-intercept

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. To find the y-intercept of our function, f(x) = (x+4)/(4x^2+4x-3), we substitute x = 0 into the function:

f(0) = (0 + 4) / (4(0)^2 + 4(0) - 3) = 4 / -3 = -4/3

Thus, the y-intercept occurs at the point (0, -4/3). This point provides a crucial anchor for visualizing the graph of the function, indicating where it crosses the vertical axis.

Delving into Asymptotes

Asymptotes are lines that the graph of a function approaches but never quite touches. They provide valuable insights into the behavior of the function as x approaches infinity or certain specific values. Rational functions can have three types of asymptotes: vertical, horizontal, and oblique (or slant).

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of the rational function equals zero, but the numerator does not. We have already identified that the denominator of f(x) = (x+4)/((2x+3)(2x-1)) equals zero at x = -3/2 and x = 1/2. Since the numerator, (x + 4), is not zero at these values, we have vertical asymptotes at x = -3/2 and x = 1/2. These vertical lines act as barriers that the graph of the function cannot cross.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To determine the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials. In our function, f(x) = (x+4)/(4x^2+4x-3), the numerator has a degree of 1 (the highest power of x is 1), and the denominator has a degree of 2 (the highest power of x is 2).

When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is the line y = 0. This means that as x becomes very large (either positive or negative), the function values approach zero. Therefore, the horizontal asymptote of f(x) is y = 0, the x-axis.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In our case, the degree of the denominator (2) is greater than the degree of the numerator (1), so there is no oblique asymptote.

Summarizing the Findings

Let's consolidate our findings about the rational function f(x) = (x+4)/(4x^2+4x-3):

  • Domain: (-∞, -3/2) ∪ (-3/2, 1/2) ∪ (1/2, ∞)
  • Y-intercept: (0, -4/3)
  • Vertical Asymptotes: x = -3/2 and x = 1/2
  • Horizontal Asymptote: y = 0
  • Oblique Asymptote: None

This comprehensive analysis provides a strong foundation for understanding the behavior and graph of the function. By identifying the domain, intercepts, and asymptotes, we gain a clear picture of how the function behaves over its entire range.

Graphing the Function

With the information gathered, we can now sketch a graph of the function f(x) = (x+4)/(4x^2+4x-3). We start by drawing the vertical asymptotes at x = -3/2 and x = 1/2, and the horizontal asymptote at y = 0. The y-intercept at (0, -4/3) gives us a point on the graph.

Knowing the asymptotes act as guidelines, we can sketch the curves of the function. The function will approach the asymptotes but never cross them. In the intervals defined by the vertical asymptotes, the function will either approach positive or negative infinity. By plotting a few additional points, we can refine the graph and ensure its accuracy.

Conclusion

Analyzing rational functions involves a systematic approach that combines algebraic techniques with graphical interpretation. By determining the domain, intercepts, and asymptotes, we can gain a thorough understanding of the function's behavior. The function f(x) = (x+4)/(4x^2+4x-3) serves as an excellent example of how these concepts come together to reveal the intricate nature of rational functions. This exploration not only enhances our mathematical toolkit but also provides valuable insights into the world of functions and their applications. Through careful analysis and visualization, we can unlock the secrets hidden within these mathematical expressions and appreciate the elegance and power of rational functions.

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