What are the vertical asymptotes of \[f(x) = \frac{x + 1}{x^2 – 2}?\] “The x-axis spans from below negative 4 to just above 4, and the y-axis spans from below negative 10 to just above 10. The x-axis has a scale of 2 in increments of 0.5, and the y-axis has a scale of 10 in increments of 2. The leftmost branch is a sharp concave curve in the third quadrant, starting from negative infinity near x = negative 1.5, increasing steeply, and then abruptly approaching the horizontal asymptote near y = 0.  The middle branch is between the asymptotes, decreasing from positive infinity near x = negative 1.5 in the second quadrant, passing through the point (negative 1,0) in a flat pattern and continuing downward past negative infinity near x = 1.5 in the fourth quadrant.   The rightmost branch is a sharp convex curve in the first quadrant, starting from positive infinity near x= 1.5 and decreasing steeply before leveling off as it approaches the horizontal asymptote near y= 0.  “

What is the x-intercept of \( f(x) = \frac{x}{x^2 – 1} \)?

Subtract the following rational expressions: \[\frac{x^2}{x + 1} – \frac{2x}{x – 1}\]

Simplify \( (4 + 2i) – (1 – 3i) \).

Subtract the rational expressions: \[\frac{x + 2}{x^2 + 3x + 2} – \frac{1}{x^2 + 3x + 2}\]

For \[f(x) = \frac{2x}{x – 3},\] what is the vertical asymptote? The x-axis spans from below zero to above 5, and the y-axis spans from below negative 10 to above 10. The x-axis has a scale of 5 in increments of 1, and the y-axis has a scale of 10 in increments of 2. The convex curve is in the first quadrant, passing through the points (3.5, 14) and (6,4). It starts from positive infinity above the vertical asymptote near x= 3. It decreases steeply before leveling off as it approaches the horizontal asymptote near y= 1. The concave curve spans the fourth, and second quadrants, passing through (2, negative 4) and the point slightly below (negative 2, 1). It starts from negative infinity below the vertical asymptote near x= 3, increasing steeply, and then approaching the horizontal asymptote near y =1 in the second quadrant after passing through the origin (0,0). 

Add the following rational expressions: \[\frac{2}{x^2 – 1} + \frac{3}{x + 1}\]

For \( y = \frac{5x}{x^2 + 2x – 3} \), what is the end behavior as \( |x| \to \infty \)? “The x-axis spans from below negative 5 to just above 5, and the y-axis spans from below negative 40 to just above 20. The x-axis has a scale of 5 in increments of 1, and the y-axis has a scale of 20 in increments of 5.  The leftmost branch is a sharp concave curve in the third quadrant, starting from negative infinity near x = negative 3, increasing steeply, and then approaching the horizontal asymptote near the negative x axis.  The middle branch is between the asymptotes, decreasing from positive infinity near x = negative 3 in the second quadrant, gradually passing through the origin (0,0) in a diagonal pattern and continuing downward past negative infinity at x = 1 in the fourth quadrant.   The rightmost branch is a sharp convex curve in the first quadrant. It starts from positive infinity near x= 1 and decreases steeply before leveling off as it approaches the horizontal asymptote near the positive x-axis.  “

Which is a factor of the equation shown in the graph?  The x-axis spans from negative 4 to beyond 2, and the y-axis spans from below negative 5 to 10. The x-axis has a scale of 2 in increments of 0.5 and the y-axis has a scale of 5 in increments of 1. The green cubic function has a local minimum at (negative 2, negative 5) and a local maximum at (0, 8). The function decreases from the top left of second quadrant, reaching its minimum, then increases to its maximum on the y-axis before decreasing again towards the fourth quadrant. The curve continues to extend out of view in both directions. 

Simplify the division: \[\frac{4x}{x^2 + 2x + 1} \div \frac{2}{x + 1}\]