Finding Zeros Of F(x) = (x+5)/(2x-3) A Step-by-Step Guide
In mathematics, finding the zeros of a function is a fundamental task. The zeros, also known as roots or x-intercepts, are the values of x for which the function f(x) equals zero. These points are crucial for understanding the behavior of the function, graphing it accurately, and solving related equations. This article will delve into the process of finding the zeros of the function f(x) = (x+5)/(2x-3), providing a step-by-step explanation and emphasizing the underlying concepts.
Understanding Zeros of a Function
Before we dive into the specific function, let's clarify what it means to find the zeros of a function. A zero of a function f(x) is a value x = a such that f(a) = 0. Graphically, these are the points where the graph of the function intersects the x-axis. For polynomial functions, the zeros directly correspond to the solutions of the equation f(x) = 0. However, for rational functions like the one we are examining, we need to consider both the numerator and the denominator.
To find the zeros, we set the function equal to zero and solve for x:
f(x) = 0
In the case of a rational function, which is a function expressed as a ratio of two polynomials, f(x) = P(x) / Q(x), the zeros occur when the numerator P(x) is equal to zero, provided that the denominator Q(x) is not simultaneously zero. This is because a fraction is only equal to zero if its numerator is zero. The denominator being zero would make the function undefined.
For the function f(x) = (x+5)/(2x-3), we identify the numerator as P(x) = x+5 and the denominator as Q(x) = 2x-3. We will first set the numerator to zero and solve for x. Then, we will check if the solution we obtain makes the denominator zero as well. If it does, then that value is not a zero of the function, but rather a point of discontinuity (a vertical asymptote). Understanding this nuanced approach is crucial for accurately determining the zeros of rational functions.
Step-by-Step Solution for f(x) = (x+5)/(2x-3)
Now, let's apply the concept to our function f(x) = (x+5)/(2x-3). We will follow a clear, step-by-step approach to find the zeros.
Step 1: Set the function equal to zero
Our first step is to set the function f(x) equal to zero. This is the foundational step in finding the zeros, as we are looking for the x values that make the function's output zero.
(x+5)/(2x-3) = 0
Step 2: Set the numerator equal to zero
As mentioned earlier, a rational function is zero only when its numerator is zero. Therefore, we set the numerator, x+5, equal to zero:
x + 5 = 0
Step 3: Solve for x
To solve for x, we subtract 5 from both sides of the equation:
x + 5 - 5 = 0 - 5
This simplifies to:
x = -5
So, we have found a potential zero of the function, x = -5. However, we need to verify that this value does not make the denominator zero.
Step 4: Check the denominator
We must ensure that the denominator, 2x - 3, is not zero when x = -5. If it is, then x = -5 is not a zero of the function but a vertical asymptote.
Substitute x = -5 into the denominator:
2(-5) - 3 = -10 - 3 = -13
Since the denominator is -13, which is not zero, x = -5 is indeed a zero of the function.
Step 5: State the zero(s) of the function
Therefore, the zero of the function f(x) = (x+5)/(2x-3) is x = -5. This means that the graph of the function intersects the x-axis at the point (-5, 0).
This step-by-step process provides a clear and concise way to find the zeros of a rational function. Understanding each step is crucial for solving similar problems and comprehending the behavior of rational functions.
Graphical Interpretation of the Zero
Understanding the graphical interpretation of the zeros of a function is essential for visualizing its behavior. As we determined, the zero of the function f(x) = (x+5)/(2x-3) is x = -5. Graphically, this corresponds to the point where the graph of the function crosses or touches the x-axis. This point is also known as the x-intercept.
When we plot the graph of f(x) = (x+5)/(2x-3), we will observe that the curve intersects the x-axis at x = -5. This confirms our algebraic solution. Additionally, the graph will exhibit a vertical asymptote at x = 3/2, which is the value that makes the denominator zero. The vertical asymptote indicates that the function approaches infinity (or negative infinity) as x approaches 3/2. The zero, on the other hand, represents a point where the function's value is precisely zero.
The graphical representation provides a visual confirmation of our algebraic solution and helps in understanding the overall behavior of the function. It highlights the significance of zeros as key features of the graph and their relationship to other important characteristics like asymptotes.
Importance of Finding Zeros
Finding the zeros of a function is not merely an academic exercise; it has significant practical applications in various fields. Zeros are essential for solving equations, modeling real-world phenomena, and understanding the behavior of systems.
In mathematics, finding the zeros is crucial for solving polynomial equations and inequalities. The zeros represent the solutions to the equation f(x) = 0, and knowing these solutions allows us to analyze the intervals where the function is positive or negative. This information is vital in optimization problems, where we seek to find the maximum or minimum values of a function.
In engineering and physics, zeros can represent equilibrium points, critical values, or points of stability in a system. For example, in a circuit, the zeros of a transfer function can indicate the frequencies at which the circuit's output is zero. In mechanics, the zeros of a potential energy function can correspond to stable equilibrium positions.
In economics, zeros can represent break-even points or points of market equilibrium. For instance, the zero of a profit function indicates the quantity of goods that must be sold to cover the costs. Similarly, in finance, zeros can represent the points where an investment yields no return.
Understanding the zeros of a function provides valuable insights into the system being modeled and allows us to make informed decisions and predictions. Therefore, mastering the techniques for finding zeros is a fundamental skill in many disciplines.
Conclusion
In conclusion, finding the zeros of a function is a critical concept in mathematics with wide-ranging applications. For the function f(x) = (x+5)/(2x-3), we have demonstrated a step-by-step approach to identify its zero, which is x = -5. This process involves setting the numerator of the rational function equal to zero and verifying that the denominator is not simultaneously zero. The zero represents the x-intercept of the function's graph and provides valuable information about its behavior.
The importance of finding zeros extends beyond mathematics into various fields, including engineering, physics, economics, and finance. Zeros represent solutions, equilibrium points, break-even points, and critical values in different contexts. By understanding how to find and interpret zeros, we gain insights into the behavior of functions and systems, enabling us to solve problems and make informed decisions.
Mastering the techniques for finding zeros is a fundamental skill for students and professionals alike. Whether dealing with polynomial, rational, or other types of functions, the concept of zeros remains a cornerstone of mathematical analysis and its applications.