Graph f(x) = (x + 3)/(x - 1). Show the vertical asymptote, horizontal asymptote, and both intercepts.

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Let's graph the rational function f(x) equals x plus 3 divided by x minus 1. First, we identify the vertical asymptote by finding where the denominator equals zero. Setting x minus 1 equal to zero, we get x equals 1. This gives us a vertical asymptote at x equals 1, shown by the red dashed line. Next, we find the horizontal asymptote. Since both numerator and denominator have the same degree, the horizontal asymptote is y equals 1, shown by the green dashed line. For the x-intercept, we set the function equal to zero, which gives us x equals negative 3. For the y-intercept, we substitute x equals 0, which gives us y equals negative 3. The function approaches negative infinity as x approaches 1 from the left, and positive infinity as x approaches 1 from the right. Now let's analyze the behavior of our function f(x) equals x plus 3 divided by x minus 1 in more detail. As x approaches 1 from the left, the denominator approaches zero from the negative side, making the function approach negative infinity. This is shown by the red arrow pointing downward. As x approaches 1 from the right, the denominator approaches zero from the positive side, making the function approach positive infinity, shown by the red arrow pointing upward. For the horizontal asymptote, as x approaches negative infinity, the function approaches 1 from below. Similarly, as x approaches positive infinity, the function approaches 1 from above. These behaviors are shown by the green arrows. The function crosses the x-axis at negative 3 and the y-axis at negative 3. This rational function creates a hyperbola with two separate branches that never cross the vertical asymptote at x equals 1.