Question 1

Given the function $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$, which of the following is the correct equation for its vertical asymptote(s)?

Accepted Answer

$$x = -10$$

Question 2

For the function $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$, what happens to $$f(x) as$$ $$x$$ approaches $$1$$?

Accepted Answer

$$f(x) is $$undefined$

Question 3

Which of the following steps is necessary to find the vertical asymptotes of $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$?

Accepted Answer

Set the denominator equal to zero and solve for $$x$$

Question 4

After factoring the denominator of $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$, which values of $$x$$ make the denominator zero?

Accepted Answer

$$x = -10$$ and $$x = 1$$

Question 5

Why is $$x = 1$$ not a vertical asymptote for $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$?

Accepted Answer

Because the numerator and denominator both become zero at $$x = 1$$, resulting in a hole, not an asymptote

Question 6

If $$f(x) = \frac{x-1}{(x+10)(x-1)}$$, what is the simplified form of $$f(x)$$ for $$x 
eq 1$$?

Accepted Answer

$$\frac{1}{x+10}$$

Question 7

Which of the following best describes the behavior of $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$ near $$x = -10$$?

Accepted Answer

$$f(x) is $$undefined and there is a vertical asymptote at $$x = -10$$

Question 8

Suppose $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$ models the profit per unit (in dollars) for a company, where $$x is $$the number of units produced. What does the vertical asymptote at $$x = -10$$ imply in this context?

Accepted Answer

Profit per unit becomes unbounded as production approaches $$-10$$ units, which is not practical

Question 9

Which of the following statements is true about the function $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$?

Accepted Answer

It has a vertical asymptote at $$x = -10$$ and a hole at $$x = 1$$

Question 10

If you were to graph $$f(x) = \frac{x-1}{x^2$ + 9x - 10}$$, what feature would you expect to see at $$x = 1$$?

Accepted Answer

A removable discontinuity (hole)

Finding Vertical Asymptotes for Rational Functions

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