Find Range of Rational Functions

Find the range of real valued rational functions using different techniques. There are also matched problems with answers at the bottom of the page.

Example 1: Find the Range of function f defined by

Solution to Example 1


  • Let us first write the given function as an equation as follows



  • Solve the above equation for x

    y (2x - 3) = x + 1

    2 y x - 3 y = x + 1

    2 y x - x = 3 y + 1

    x (2 y - 1) = 3 y + 1

    x = (3 y + 1) / (2 y - 1)

  • The above expression of x in terms of y shows that x is real for all real values of y except 1/2 since y = 1 / 2 will make the denominator 2 y - 1 = 0. Hence the range of f, which is the set of all possible values of y, is given by

    (-∞ , 1 / 2) ∪ (1 / 2 , +∞)

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 1: Find the range of function f defined by

Answer at the bottom of the page.




Example 2: Find the Range of function f defined by

Solution to Example 2

  • Write the given function as an equation

  • Rewrite the above equation for x in standard form and solve using quadratic formulae

    y x2 - x - 9 y - 2 = 0

  • Find the discriminant to the above equation

    Δ = (-1)2 - 4 y (-9 y - 2) = 36 y2 + 8 y + 1

  • Using the quadratic formulas, the above equation gives the solutions



  • The solutions x1,2 are real if 36 y2 + 8 y + 1 ≥ 0 and y ≠0. Hence we need to solve the inequality

    36 y2 + 8 y + 1 ≥ 0

  • The discriminant of 36 y2 + 8 y + 1 is equal to

    82 - 4(36)(1) = - 80

  • Since the discriminant is negative, one test value shows that 36 y2 + 8 y + 1 is always positive. For y = 0, we need to set y= 0 in the equation y = (x + 2) / (x2 - 9) and solve it

    (x + 2) / (x2 - 9) = 0

  • gives solution x = -2 and therefore y = 0 is also in the range of f. Hence the range of f is given by

    ( -∞ , +∞ )

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 2: Find the range of function f defined by

Answer at the bottom of the page.




Example 3: Find the Range of function f defined by

Solution to Example 3

  • Write the above function as an equation.


  • Rewrite the above as follows.

    y x2 - x + y - 2 = 0

  • Solve for x using the quadratic formulae.



  • The above solutions are real if the radicand is not negative and y not equal to 0. Hence we need to solve the inequality

    1 - 4 y2 + 8y ≥ 0

  • The solution set to the above inequality is

    1 - √5 / 2 ≤ y ≤ 1 + √5 / 2 with y = 0 excluded.

  • But if we set y to 0 in the first equation, we obtain

    0 = (x + 2) / (x2 + 1)

  • which gives x = - 2 and hence y = 0 is also included in the range of f for now. Hence the range of f is given by the interval

    [ 1 - √5 / 2 , 1 + √5 / 2 ]

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 3: Find the range of function f defined by

Answer at the bottom of the page.




Example 4: Find the Range of function f defined by

Solution to Example 4

  • Write the function as an equation


  • rewrite as a quadratic equation in x

    x2( y - 1 ) = 2 - y

  • Solve for x



  • The above solutions are real if the radicand is not negative. Hence we need to solve the inequality

    (2 - y) / (y - 1) ≥ 0

  • whose solution set is given by

    1 < y ≤ 2

  • The range of the given function is given by the interval

    (1 , 2]

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 4: Find the range of function f defined by

Answer at the bottom of the page.




Example 5: Find the Range of function f defined by

Solution to Example 5

  • Write the function as an equation


  • rewrite as a quadratic equation in x

    x2y = y + 1

  • Solve for x



  • for the solutions to be real, the radicand must be non negative. Hence

    (y + 1) / y ≥ 0

  • The solution set to the above inequality is

    (-∞ -1 ] ∪ (0 , +∞)

  • which is also the range of the given function.

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 5: Find the range of function f defined by

Answer at the bottom of the page.




Example 6: Find the Range of function f defined by

Solution to Example 6

  • Write the function as an equation


  • rewrite the above as a quadratic function
    2 x 2 - y x - y - 1 = 0

  • solve for x



  • the above solutions are real if the radicand is not negative. Hence we need to solve the inequality

    y2 + 8 y + 8 ≥ 0

  • whose solution set is given by the interval

    (-∞ , - 4 - 2√2 ] ∪ [- 4 + 2√2 , +∞)

  • which is also the range of the given function

    See graph below of function f given above and compare range found and that of the graph.

Matched Problem 6: Find the range of function f defined by

Answer at the bottom of the page.




Answers to matched problems

1. (-∞ , -1) ∪ (-1 , +∞)

2. (-∞ , +∞)

3. [ (2 - √10) /-12 , (2 + √10) /-12 ]

4. (1/2 , 5]

5. (-∞ 0 ) ∪ [ 3/4 , +∞)

6. (-∞ , 4-2√6 ] ∪ [4+2√6 , +∞)

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