Finding X-Intercepts Of F(x)=(5x^2-25x)/x A Step-by-Step Guide
The question at hand involves finding the -intercept(s) of the function . The -intercepts, also known as the roots or zeros of a function, are the points where the graph of the function intersects the -axis. At these points, the value of the function, , is equal to zero. To find these intercepts, we need to solve the equation .
To solve for the -intercepts of the given rational function, we first set the function equal to zero:
A rational function is equal to zero when its numerator is equal to zero (and the denominator is not zero, as this would make the function undefined). Therefore, we focus on the numerator:
Now, we can factor out the common factor, which is :
This equation is satisfied when either or . Solving these two equations gives us:
So, we have two potential -intercepts: and . However, we must check for any values of that would make the original function undefined. A rational function is undefined when its denominator is equal to zero. In this case, the denominator is . Thus, the function is undefined when .
Since makes the denominator zero, it is not a valid -intercept. It represents a point of discontinuity (specifically, a removable singularity or a hole) in the graph of the function. Therefore, we discard as an -intercept.
The only remaining potential -intercept is . Since this value does not make the denominator zero, it is a valid -intercept. Thus, the function has only one -intercept, which is .
Therefore, the correct answer is:
A.
In summary, finding the -intercepts of a rational function involves setting the function equal to zero, solving for , and then checking for any values of that make the denominator zero. These values must be excluded from the set of -intercepts.
Understanding X-Intercepts and Rational Functions
When dealing with rational functions, identifying the x-intercepts is a crucial step in understanding the function's behavior and graph. X-intercepts, also known as roots or zeros, are the points where the graph of the function intersects the x-axis. At these points, the function's value, denoted as f(x), is equal to zero. In simpler terms, they are the solutions to the equation f(x) = 0. For rational functions, which are essentially fractions where the numerator and denominator are polynomials, the process of finding x-intercepts requires careful consideration of both the numerator and the denominator.
To find the x-intercepts of a rational function, the primary focus is on the numerator. A rational function equals zero when its numerator is zero, provided the denominator is not simultaneously zero. This condition is vital because a zero denominator would render the function undefined. Therefore, the initial step involves setting the numerator of the rational function equal to zero and solving for the variable, typically x. The solutions obtained are potential x-intercepts, but they must be validated against the denominator.
After finding the potential x-intercepts from the numerator, it is essential to check whether these values also make the denominator zero. If a potential x-intercept makes the denominator zero, it is not a valid x-intercept. This is because division by zero is undefined in mathematics, and such a value would represent a point of discontinuity in the graph of the function, rather than an x-intercept. These points of discontinuity can manifest as vertical asymptotes or holes in the graph, depending on whether the factor causing the zero in the denominator can be canceled with a factor in the numerator.
Consider the function f(x) = (x - 2) / (x + 3). To find the x-intercepts, we set the numerator x - 2 equal to zero, which gives us x = 2. We then check if this value makes the denominator zero. Substituting x = 2 into the denominator x + 3 yields 5, which is not zero. Therefore, x = 2 is a valid x-intercept. However, if we consider the function g(x) = (x - 2) / (x - 2), setting the numerator x - 2 equal to zero gives us x = 2. But this value also makes the denominator zero, meaning x = 2 is not an x-intercept but rather a hole in the graph at x = 2.
The process of identifying x-intercepts in rational functions is thus a two-step process: first, find potential x-intercepts by setting the numerator to zero, and second, validate these by ensuring they do not simultaneously make the denominator zero. This careful approach ensures accurate identification of the x-intercepts and a proper understanding of the rational function's behavior. In graphical terms, x-intercepts provide critical points for sketching the graph, indicating where the function crosses or touches the x-axis. They also play a significant role in analyzing the function's domain, range, and overall characteristics.
Step-by-Step Solution to Finding X-Intercepts
To methodically determine the x-intercepts of the function f(x) = (5x^2 - 25x) / x, a structured approach is necessary. This involves a series of steps, each crucial for arriving at the correct solution. The first step is to set the function equal to zero. This is based on the fundamental understanding that x-intercepts occur where the function's value is zero, indicating the points where the graph crosses the x-axis. By setting f(x) = 0, we establish the equation that needs to be solved to find these points.
The equation derived from setting the function to zero is (5x^2 - 25x) / x = 0. Since we are dealing with a rational function, this equation holds true when the numerator is zero, provided the denominator is not simultaneously zero. The rationale behind this is that a fraction equals zero only if its numerator is zero. However, if the denominator is also zero, the function becomes undefined, which is a critical consideration in the context of rational functions. Therefore, the next step involves focusing on the numerator of the function. We set the numerator, 5x^2 - 25x, equal to zero, which simplifies the problem to solving a polynomial equation.
The equation 5x^2 - 25x = 0 is a quadratic equation, but it can be solved more easily by factoring. Factoring involves expressing the quadratic expression as a product of simpler expressions. In this case, we identify that both terms in the equation have a common factor of 5x. Factoring out 5x from the expression yields 5x(x - 5) = 0. This factored form is incredibly useful because it transforms the problem of solving a quadratic equation into the simpler task of finding the values of x that make each factor equal to zero.
With the equation in the factored form 5x(x - 5) = 0, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This leads to two separate equations: 5x = 0 and x - 5 = 0. Solving the first equation, 5x = 0, involves dividing both sides by 5, which gives us x = 0. Solving the second equation, x - 5 = 0, involves adding 5 to both sides, which gives us x = 5. These two values, x = 0 and x = 5, are the potential x-intercepts of the function.
However, it is crucial to validate these potential x-intercepts by checking them against the original function, particularly the denominator. Rational functions are undefined when the denominator is zero, as division by zero is not allowed in mathematics. In the given function f(x) = (5x^2 - 25x) / x, the denominator is simply x. If we substitute x = 0 into the denominator, we get zero, which means that x = 0 makes the function undefined. Therefore, x = 0 is not a valid x-intercept. On the other hand, substituting x = 5 into the denominator gives us 5, which is not zero, so x = 5 remains a valid x-intercept.
Common Mistakes and How to Avoid Them
Finding the x-intercepts of rational functions involves a series of steps where errors can easily occur if not approached carefully. One of the most common mistakes is overlooking the importance of the denominator. When solving for x-intercepts, it's tempting to focus solely on the numerator, setting it equal to zero and solving for x. However, this approach can lead to incorrect results if the solutions obtained also make the denominator zero. A rational function is undefined when its denominator is zero, so any value of x that makes the denominator zero cannot be a valid x-intercept. These values represent points of discontinuity, such as vertical asymptotes or holes, rather than x-intercepts. To avoid this mistake, always check potential x-intercepts against the denominator to ensure they don't make it zero.
Another frequent error is failing to factor the numerator and denominator completely before simplifying or solving. Factoring is a critical step in finding x-intercepts because it helps identify common factors that can be canceled out. However, if factoring is not done completely, potential x-intercepts might be missed, or extraneous solutions might be included. For instance, consider a function where both the numerator and denominator have a common factor of (x - 2). If this factor is not identified and canceled, the value x = 2 might incorrectly be identified as both an x-intercept and a point of discontinuity. Complete factoring ensures that all common factors are identified and canceled, leading to a simplified function that accurately represents the x-intercepts and discontinuities.
Sign errors in algebraic manipulation are also a common pitfall when finding x-intercepts. These errors can occur during various steps, such as distributing negative signs, combining like terms, or solving equations. A simple sign error can lead to an incorrect solution, thereby misidentifying the x-intercepts. For example, when solving the equation x - 5 = 0, a sign error might lead to the incorrect solution x = -5 instead of the correct solution x = 5. To minimize sign errors, it's crucial to double-check each step of the algebraic manipulation, pay close attention to the signs of the terms, and use parentheses when necessary to avoid confusion.
Additionally, overlooking the zero-product property is another mistake that can occur when solving factored equations. The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is essential for solving equations in factored form. However, if this property is not applied correctly, some x-intercepts might be missed. For example, if an equation is factored as x(x - 3) = 0, the zero-product property tells us that either x = 0 or x - 3 = 0. Solving these equations gives us the x-intercepts x = 0 and x = 3. Failing to recognize both factors can lead to an incomplete solution.
To avoid these common mistakes, a systematic approach is crucial. This includes carefully factoring the numerator and denominator, checking potential x-intercepts against the denominator, double-checking algebraic manipulations for sign errors, and correctly applying the zero-product property. By adopting these strategies, the accuracy of finding x-intercepts in rational functions can be significantly improved.
Real-World Applications of X-Intercepts
X-intercepts, while a fundamental concept in mathematics, have a wide array of practical applications across various real-world scenarios. Understanding and calculating x-intercepts can provide valuable insights in fields ranging from physics and engineering to economics and business. These points, where a function's graph intersects the x-axis, represent critical values where the dependent variable (often denoted as y or f(x)) equals zero, making them significant for analysis and decision-making.
In physics, x-intercepts are often used to determine equilibrium points in systems. For example, in mechanics, the position of an object under the influence of forces can be modeled as a function of time. The x-intercepts of this function represent the times at which the object's position is zero, indicating moments when the object is at a reference point or has returned to its initial position. Similarly, in electrical circuits, x-intercepts can represent the times when the voltage or current is zero, which can be crucial for analyzing circuit behavior and identifying potential issues.
Engineering disciplines also heavily rely on x-intercepts for various applications. In structural engineering, for instance, the deflection of a beam under load can be modeled as a function of the applied load. The x-intercepts of this function indicate the load values at which the beam experiences no deflection, which is a critical consideration in ensuring structural integrity and safety. In control systems engineering, x-intercepts can represent the points at which a system's output reaches a desired target value, allowing engineers to design controllers that effectively regulate system behavior.
In economics and finance, x-intercepts play a crucial role in break-even analysis. A break-even point is the point at which total costs equal total revenue, resulting in zero profit or loss. By modeling costs and revenues as functions of production quantity or sales volume, the x-intercepts of the profit function (revenue minus costs) represent the break-even points. These points are essential for businesses to determine the minimum level of sales or production required to cover their costs and start generating a profit. X-intercepts can also be used to analyze market equilibrium, where supply and demand curves intersect, indicating the price and quantity at which the market is in balance.
Business and marketing also utilize the concept of x-intercepts in various analytical models. For example, in marketing, the response of customers to an advertising campaign can be modeled as a function of advertising expenditure. The x-intercepts of this function may indicate the points at which the campaign generates no additional sales or leads, helping marketers optimize their advertising budgets and strategies. In project management, x-intercepts can represent milestones or deadlines at which specific project deliverables are expected to be completed, providing a benchmark for tracking progress and ensuring timely project completion.
In environmental science, x-intercepts can be used to model and analyze environmental phenomena. For instance, the concentration of a pollutant in a body of water can be modeled as a function of time or distance from the source of pollution. The x-intercepts of this function indicate the points at which the pollutant concentration is zero, which can be crucial for assessing the extent of pollution and implementing remediation measures. Similarly, in climate modeling, x-intercepts can represent the points at which certain climate variables, such as temperature or sea level, reach critical thresholds, informing climate change mitigation and adaptation strategies.
These diverse applications highlight the versatility and significance of x-intercepts as a mathematical tool. By understanding and applying the concept of x-intercepts, professionals in various fields can gain valuable insights, make informed decisions, and solve real-world problems effectively.
Conclusion
In conclusion, finding the x-intercepts of the function f(x) = (5x^2 - 25x) / x requires a careful and methodical approach. The process involves setting the function equal to zero, solving for x, and, most importantly, checking for values that make the denominator zero. In this specific case, while the initial solution process yields x = 0 and x = 5 as potential x-intercepts, the value x = 0 is invalid because it makes the denominator zero, rendering the function undefined. Therefore, the only valid x-intercept for the given function is x = 5. This detailed analysis underscores the importance of considering both the numerator and the denominator when working with rational functions to accurately determine their x-intercepts.
