On this page you will find a focused collection of rational function math problems along with video solutions!
Visit this page directly at hunkim.com/rational
- Basic rational function graph:
f(x)=\frac{1}{x}- Sketch this function and label 3 points
- Domain?
- Range?
- Equation of the asymptotes?
- Non-permissible values for x
- The two big no-no’s in math:
f(x)=\frac{\sqrt{x-2}}{x^2-9}- What are the non-permissible values?
- Domain?
- Basic rational function transformations:
f(x)=\frac{1}{x-a}+b. What happens to the graph as a and b changes? - Simplify the following rational expression using factoring: \frac{{3x - 6}}{{2{x^2} + x - 10}}
- Don’t forget to pull out the -1:
Simplify\frac{1-x}{x^2-1} - Recognize difference of squares involving radicals:
Simplify \frac{x^2-7}{x-\sqrt{7}} - Multiply rational functions:
\frac{{{x^2} - x - 12}}{{{x^2} - 9}} \times \frac{{{x^2} - 4x + 3}}{{{x^2} - 4x}} - Divide rational functions:
\frac{{{x^2} - 4}}{{{x^2} - 4x}} \div \frac{{{x^2} + x - 6}}{{{x^2} + x - 20}}- Simplify
- Domain? (remember restriction on every step!)
- Mixing multiplying and dividing:
Simplify \frac{{2{x^2} - 7x - 15}}{{2{x^2} - 10x}} \div \frac{{4{x^2} - 9}}{6} \times (3 - 2x) - Basic rectangle area problem:
The area of a rectangle is x^2-9. The length of one side is \frac{x^2-2x-3}{x+1}. Find the height of the rectangle. - Subtracting rational functions with the same denominator:
Simplify \frac{2x}{y}-\frac{x-1}{y} - Adding rational functions with different denominators:
Simplify \frac{2x}{xy}+\frac{4}{x^2} - Find a common denominator by factoring:
Simplify \frac{a^2-20}{a^2-4}+\frac{a-2}{a+2} - Simplify a rational expression with layers of fractions:
Simplify \frac{1+\frac{1}{x}}{x-\frac{1}{x}} - Solve the rational equation you could cross multiply):
\frac{2x+1}{x-4}=\frac{x-3}{x+1} - Solve the rational equation involving addition:
\frac{2}{x-2}+\frac{1}{x}=-1 - Solve the equation involving three rational expressions:
\frac{2}{{{x^2} - 4}} + \frac{{10}}{{6x + 12}} = \frac{1}{{x - 2}} - Solve the rational equation involving extraneous roots:
\frac{{4x - 1}}{{x + 2}} - \frac{{x + 1}}{{x - 2}} = \frac{{{x^2} - 4x + 24}}{{{x^2} - 4}} - What the half way point between:
- 2\frac{2}{3} and \frac{17}{4}
- Challenge: \frac{3}{a} and \frac{7}{2a}
- 2\frac{2}{3} and \frac{17}{4}
- Challenge: An image found by a convex lens is described by the equation \frac{1}{f}=\frac{1}{u}+\frac{1}{v}. Find f
- Challenge: Simplify \left( {\frac{p}{{p - x}} + \frac{q}{{q - x}} + \frac{r}{{r - x}}} \right) - \left( {\frac{x}{{p - x}} + \frac{x}{{q - x}} + \frac{x}{{r - x}}} \right)
- Challenge: Given b = \frac{1}{a} and \frac{{\frac{1}{a} - \frac{1}{b}}}{{\frac{1}{a} + \frac{1}{b}}} = \frac{4}{5}, solve a.
Rational Word Problems
- The sum of a number and twice the reciprocal is \frac{9}{2}. Find the number.
- Find two consecutive even integers whose reciprocals sum to be \frac{11}{60}.
- Tap A fills the tub in 4 hours. Tab B fills the tub in 2 hours. How long does it take to fill the tub when tap A and B work together?
- A cold water tap can fill a tub in 6 minutes, and a hot water tap can fill the tub in 8 minutes. A drain can empty a full tub in 10 minutes. If both taps are on and the drain is open, how long will it take to fill the tub?
- You travel from point A to B and the distance is 70 km. On he way back you are 6 kph slower because you are tired. The total time for the trip was 8\frac{1}{2} hours. What was your average speed from point A to B?
- You travel 120 km to Whistler by car, and then return by bus. The average speed of the car is 15 km/h greater than the average speed of the bus. What is the correct simplified expression for the total time of your trip?
- A boat travels 40 km downstream in the same time it takes to travel 30 km upstream. If the current flows at 6 kph, what is the speed of the boat in still water?
- On your first six tests you average a score of 36/50. What average mark must you receive on the next four tests so that your average is 80% in the course?
