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

What is the domain of \[f(x) = \frac{x}{x^2 – 9}?\]

Identify the zeros and end behavior of the graph. The x-axis spans from negative 5 to 5, and the y-axis spans from below negative 5 to just above 5. Both axes have a scale of 5 in increments of 1. The red curve represents a cubic polynomial function with two turning points. It starts from negative infinity in the third quadrant, increases to a local maximum slightly below (negative 2.5, 1), then decreases to a local minimum at a point slightly left of (0, negative 6), and finally rises steeply towards positive infinity in the first quadrant. The curve crosses the x-axis at the points (negative 3, 0), (negative 2, 0), and (1, 0), while crossing the y-axis at (0, negative 6).

The zeros of \( f(x) = \frac{x}{x^2 – 4} \) are:

What happens to the graph of \( f(x) = \frac{1}{x} \) when it is replaced with \( f(x) = \frac{1}{2x} \)? The x-axis spans from below negative 2 to above 2, and the y-axis spans from below negative 5 to above 5. 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 convex curve spans the first quadrant, passing through the points (0.25, 2) and (1, 0.5). It starts from positive infinity near the vertical asymptote at x = 0 and decreasing toward the horizontal asymptote at y = 0. The concave curve is in the third quadrant, passing through the points (negative 1, negative 0.5) and (negative 0.25, negative 2). It approaches negative infinity as it nears the vertical asymptote at x = 0 and levels out toward the horizontal asymptote near y= 0 as x moves left. 

Which of the following rational functions has a vertical asymptote at \( x = 5 \)?

Simplify the sum: \[\frac{x + 1}{x + 4} + \frac{x – 1}{x + 4}\]