Question

Graph the function. Describe the domain. $$y=\frac{-2 x+11}{x-5}$$

Short answer

The domain of the function \(y=\frac{-2x+11}{x-5}\) is all real numbers except 5, or in interval notation, (-∞, 5) U (5, ∞). The function is graphed as a curve slanting down to the right, with a vertical asymptote at x = 5 and a y-intercept at -2.2.

Step-by-step solution

Determine the Value That Makes the Denominator Zero

First, solve 'x - 5 = 0' to find out the value for x that makes the denominator zero. So, x = 5 is the value that makes the denominator zero.

Describe the Domain

The domain of the function is all real numbers except the value that makes the denominator zero. The value from the previous step is x = 5, so the domain of the function is all real numbers except 5. In interval notation, this can be written as (-∞, 5) U (5, ∞).

Graph the Function

To graph the function, first plot a point on the y-axis where x = 0. This will be the y-intercept. For the function \(y=\frac{-2x+11}{x-5}\), when x = 0, y = -2.2. Then plot a vertical asymptote at x = 5 and draw the curve of the function respecting the asymptote and going through the y-intercept, slanting down to the right.