Practice Set 4
10 Mixed Rational Questions
This mixed set blends restrictions, asymptotes, graph reading, and equation solving in one review round.
Practice Questions
Mixed Practice Set A: Rational Functions
Use this set to switch between graph features, equation solving, and quick interpretation so your rational-function skills stay flexible.
Question 1
What is the domain of (x - 5) / (x² - 25)?
Desmos Move: Factor the denominator first. Even if one factor cancels, both original denominator zeros still matter for the domain.
Question 2
Which feature tells you a rational function has a hole instead of a vertical asymptote?
Desmos Move: A hole usually shows up as one missing point on the simplified graph, while an asymptote changes how the graph behaves near a line.
Question 3
Solve (x - 4) / (x + 1) = 2.
Desmos Move: Multiply both sides by x + 1, solve the linear equation, and then check that the answer does not make the denominator 0.
Question 4
The graph shown has a vertical asymptote at x = -2 and an x-intercept at x = 3. Which equation fits best?
Use the asymptote and intercept together to choose the correct equation.
Desmos Move: Use the denominator for the vertical asymptote and the numerator for the x-intercept.
Question 5
For (x² + x - 6) / (x + 3), what is the y-value of the hole?
Desmos Move: Factor the numerator, simplify, and then substitute the excluded x-value into the simplified expression to find the hole.
Question 6
What is the horizontal asymptote of y = (x² + 1) / (2x² - 3)?
Desmos Move: Since the degrees match, compare the leading coefficients to find the horizontal asymptote quickly.
Question 7
What is the solution set of (x + 1) / (x + 1) = 2?
Desmos Move: The left side simplifies to 1 for every allowed x-value, so compare that simplified value to the right side.
Question 8
A delivery company models fuel cost per package as 180 / x, where x is the number of packages. What happens to the cost per package as x increases?
Desmos Move: In Desmos, make a table for several positive x-values and watch how the outputs change as the denominator grows.
Question 9
What is the y-intercept of y = 2 / (x - 4) + 1?
Desmos Move: Substitute x = 0 to find the y-intercept, then check the point on the graph if you want a quick confirmation.
Question 10
Which function has branches in Quadrants I and III with asymptotes x = 0 and y = 0?
Desmos Move: The sign controls whether the branches land in Quadrants I and III or Quadrants II and IV.