4. If g(x)=(2x−1)/(x+3), calculate the domain of the function. Identify any vertical asymptotes and horizontal asymptotes

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Let's analyze the rational function g of x equals two x minus one divided by x plus three. We need to find its domain and identify any vertical and horizontal asymptotes. First, let's determine the domain. For a rational function, the domain includes all real numbers except where the denominator equals zero. Setting x plus three equal to zero, we get x equals negative three. Therefore, the domain is all real numbers except negative three, which we can write as the union of negative infinity to negative three and negative three to positive infinity. Next, let's identify the vertical asymptote. A vertical asymptote occurs where the denominator equals zero but the numerator doesn't. We already found that the denominator equals zero at x equals negative three. Let's check the numerator at this value. When x equals negative three, the numerator equals two times negative three minus one, which is negative seven. Since negative seven is not zero, we have a vertical asymptote at x equals negative three. Finally, let's find the horizontal asymptote. For rational functions, we compare the degrees of the numerator and denominator. Both the numerator, two x minus one, and the denominator, x plus three, have degree one. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is two, and the leading coefficient of the denominator is one. Therefore, the horizontal asymptote is y equals two. To summarize, the domain of g of x equals two x minus one divided by x plus three is all real numbers except negative three, written as the union of negative infinity to negative three and negative three to positive infinity. The function has a vertical asymptote at x equals negative three and a horizontal asymptote at y equals two. Let's explore the domain of our rational function g of x equals two x minus one divided by x plus three in more detail. For any rational function, the domain consists of all real numbers except where the denominator equals zero. Step one is to identify where the denominator equals zero. Setting x plus three equal to zero, we solve for x and get x equals negative three. This is the only value we need to exclude from the domain. Step two is to exclude this value from the domain. The domain is all real numbers except negative three, which we can write as the set of real numbers minus the singleton set containing negative three. Alternatively, we can express this as the union of two intervals: from negative infinity to negative three, and from negative three to positive infinity. Let's visualize this by moving a point along the function. As we approach x equals negative three from the left, the function values decrease without bound, approaching negative infinity. As we approach from the right, the function values increase without bound, approaching positive infinity. At exactly x equals negative three, the function is undefined. The domain is visualized here as the blue segments on the x-axis. It includes all values to the left of negative three and all values to the right of negative three, but not negative three itself. This is precisely the domain we found algebraically: the union of negative infinity to negative three and negative three to positive infinity. Now let's examine the vertical asymptote of our function g of x equals two x minus one divided by x plus three. A vertical asymptote occurs at a value where the function approaches infinity as we get closer to that value. Step one is to find where the denominator equals zero. We've already determined that the denominator x plus three equals zero when x equals negative three. Step two is to check if the numerator is also zero at this point. If both the numerator and denominator are zero at the same point, we might have a removable discontinuity instead of a vertical asymptote. Let's evaluate the numerator at x equals negative three. Two times negative three minus one equals negative six minus one, which equals negative seven. Since negative seven is not zero, the numerator is non-zero when the denominator is zero. Step three is to confirm the vertical asymptote by examining the behavior of the function as x approaches negative three from both sides. Let's visualize this by moving our point along the function. As x approaches negative three from the left, that is, from values less than negative three, the function values decrease without bound, approaching negative infinity. We can see this as our point moves closer to x equals negative three from the left side. As x approaches negative three from the right, that is, from values greater than negative three, the function values increase without bound, approaching positive infinity. We can see this as our point moves closer to x equals negative three from the right side. This behavior confirms that x equals negative three is indeed a vertical asymptote of our function. The function approaches negative infinity as x approaches negative three from the left, and it approaches positive infinity as x approaches negative three from the right. Now let's examine the horizontal asymptote of our function g of x equals two x minus one divided by x plus three. A horizontal asymptote is a horizontal line that the graph of the function approaches as x approaches positive or negative infinity. Step one is to compare the degrees of the numerator and denominator. The numerator is two x minus one, which has degree one. The denominator is x plus three, which also has degree one. When the degrees are equal, we know that a horizontal asymptote exists, and it's determined by the ratio of the leading coefficients. Step two is to find the ratio of the leading coefficients. The leading coefficient of the numerator is two, and the leading coefficient of the denominator is one. The ratio is two divided by one, which equals two. Step three is to identify the horizontal asymptote. Since the ratio of the leading coefficients is two, the horizontal asymptote is y equals two. This means that as x approaches positive or negative infinity, the function values approach two. Let's visualize this by moving our point along the function. As x increases to very large positive values, we can see that the function values get closer and closer to two. Similarly, as x decreases to very large negative values, far to the left of the vertical asymptote, the function values also approach two. This confirms that y equals two is indeed the horizontal asymptote of our function. No matter how far we go to the left or right, the function values will get arbitrarily close to two but never quite reach it. This is a key characteristic of rational functions where the numerator and denominator have the same degree. Let's summarize what we've learned about the rational function g of x equals two x minus one divided by x plus three. The domain of this function is all real numbers except x equals negative three, which we can write as the union of negative infinity to negative three and negative three to infinity. The function has a vertical asymptote at x equals negative three, which occurs where the denominator equals zero. It also has a horizontal asymptote at y equals two, which is determined by the ratio of the leading coefficients of the numerator and denominator. These asymptotes help us understand the overall behavior of the function, particularly as x approaches negative three or as x approaches positive or negative infinity.