Rational Functions: Finding G(x) With Shared Intercepts

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Let's dive into the fascinating world of rational functions and explore how different transformations can affect their properties, specifically their domains and x-intercepts. We'll analyze a scenario where two rational functions, f(x) and g(x), share the same domain and have a single x-intercept at x = -10. Our goal is to determine which equation could represent g(x) in terms of f(x) from a given set of options. This problem touches upon key concepts such as function transformations, intercepts, and the behavior of rational functions. So, let's get started!

Understanding the Problem

The problem states that we have two rational functions, f(x) and g(x), that have the same domain. Remember, the domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero. Also, both functions have a single x-intercept at x = -10. An x-intercept is the point where the graph of the function crosses the x-axis, meaning the function's value is zero at that point. In other words, f(-10) = 0 and g(-10) = 0. The question asks us to identify which of the provided equations could correctly define g(x) based on f(x), considering these shared properties. We need to carefully examine each option and see if it preserves both the domain and the x-intercept.

Analyzing the Options

Let's go through each option step by step and check if it satisfies the given conditions. This is where we'll apply our understanding of function transformations and how they impact key features like domain and x-intercepts.

Option A: g(x) = 10f(x)

This option suggests that g(x) is simply a vertical stretch of f(x) by a factor of 10. A vertical stretch multiplies the output of the function by a constant. Let's analyze how this transformation affects the domain and x-intercept.

  • Domain: Multiplying a function by a constant does not change its domain. If f(x) is undefined at a certain point, then 10f(x) will also be undefined at that same point. Therefore, the domain of g(x) will be the same as the domain of f(x).
  • X-intercept: If f(-10) = 0, then g(-10) = 10 * f(-10) = 10 * 0 = 0. So, g(x) also has an x-intercept at x = -10. Furthermore, since we are simply scaling the function, any other x-intercepts of f(x) would also be x-intercepts of g(x). Since f(x) has only one x-intercept at x = -10, g(x) will also have only one x-intercept at x = -10.

Therefore, g(x) = 10f(x) satisfies both conditions: the same domain and a single x-intercept at x = -10. This looks like a promising candidate!

Option B: g(x) = f(x + 10)

This option represents a horizontal translation of f(x). Specifically, it's a shift to the left by 10 units. Remember that f(x + c) shifts the graph of f(x) horizontally by c units to the left if c is positive and to the right if c is negative. Let's examine how this affects the domain and x-intercept.

  • Domain: A horizontal translation can affect the domain of a rational function. If f(x) has a vertical asymptote at x = a, then f(x + 10) will have a vertical asymptote at x = a - 10. Thus, the domain shifts. For example, if f(x) = 1/x, then the domain is all real numbers except 0. But g(x) = f(x+10) = 1/(x+10), and the domain of g(x) is all real numbers except -10. Since we require the domains to be the same, this option isn't viable.
  • X-intercept: To find the x-intercept of g(x) = f(x + 10), we need to find the value of x for which g(x) = 0. That means f(x + 10) = 0. We know that f(x) = 0 when x = -10. Therefore, f(x + 10) = 0 when x + 10 = -10, which means x = -20. Thus, g(x) has an x-intercept at x = -20, not at x = -10.

Because the x-intercept is not at x = -10 and the domain is not necessarily the same, this option does not satisfy the conditions.

Option C: g(x) = f(x) + 10

This option represents a vertical translation of f(x), shifting the graph up by 10 units. Let's see how this transformation impacts the domain and x-intercept.

  • Domain: A vertical translation does not change the domain of a function. If f(x) is undefined at a certain point, then f(x) + 10 will also be undefined at that same point. Therefore, the domain of g(x) will be the same as the domain of f(x).
  • X-intercept: If f(-10) = 0, then g(-10) = f(-10) + 10 = 0 + 10 = 10. So, g(x) does not have an x-intercept at x = -10. Instead, it has a y-intercept of 10 at x = -10.

Since the x-intercept is not preserved, this option does not satisfy the given conditions.

Conclusion

After carefully analyzing each option, we found that only option A, g(x) = 10f(x), satisfies both conditions: it maintains the same domain as f(x) and has a single x-intercept at x = -10. Therefore, the correct answer is:

A. g(x) = 10f(x)

In summary, to solve this problem, we considered the definitions of rational functions, domains, and x-intercepts. We investigated each potential answer using our understanding of function transformations, particularly focusing on how vertical stretches, horizontal translations, and vertical translations affect these key properties. By methodically checking each condition, we were able to identify the correct equation that represents g(x) in terms of f(x). Remember to always consider the core definitions and the impact of transformations when dealing with functions!