Investigating rational functions · Grade 11 · 4 units
In Investigating rational functions (Grade 11 · 4 units) you practise 6 skills: The domain · The horizontal asymptote · Inequalities and where it is positive or negative · Extremum points · Even and odd functions · Finding a parameter. Investigating a rational function starts with its domain: the function is not defined where the denominator is 0, and its graph has a vertical asymptote there. The programme has no functions whose numerator and denominator are 0 at the same point. Far from the origin the highest powers decide: when the degree of the numerator is lower than the degree of the denominator, the horizontal asymptote is the horizontal axis, and when the degrees are equal, it is at the height of the quotient of the coefficients of the highest powers.
The sign of a rational function changes only where the numerator or the denominator is 0. Where the numerator is 0 the graph meets the horizontal axis, and it meets the vertical axis at the value of the function at 0. Mark those places on the number line and check the sign in each interval between them: that is also how inequalities are solved. A place where the denominator is 0 is never part of the solution, because the function is not defined there. Turning points come from the derivative: where it is 0 and changes sign there is a turning point, and on a closed interval the values at the ends are compared too. An even function takes the same value at a number and at its negative, and its graph is symmetric about the vertical axis. An odd function takes opposite values there, and its graph is symmetric about the origin.
Tip: Before any question, find where the numerator and the denominator are 0: every sign chart is built from those places.
What you learn
The domain
For example: Here is the function f(x) = (5x − 7) / (x + 5) . For which value of x is it not defined?
Answer and solution
The answer: −5
The denominator x + 5 is 0 when x = −5 , and the function is not defined there.
The horizontal asymptote
For example: The graph of y = (−7x − 7) / (6x + 7) has a horizontal asymptote y = k . Find k as a fraction in its simplest form.
Answer and solution
The answer: −7/6
The numerator and the denominator both have degree 1, so the function approaches the quotient of the coefficients of x : k = −7/6 .
Inequalities and where it is positive or negative
For example: For which values of x does the inequality (x + 5) / (x − 6) > 0 hold?
- −5 < x < 6
- x > 6
- x < −5
- x < −5 or x > 6
Answer and solution
The answer: x < −5 or x > 6
The numerator is 0 when x = −5 and the denominator when x = 6 . To the left of both values and to the right of both, the numerator and the denominator have the same sign, so the function is positive there, and between them it is negative. So the answer is x < −5 or x > 6 .
Extremum points
For example: Here is the function f(x) = (x² + 1) / x . Find its minimum point on the interval x > 0 .
Answer and solution
The answer: (1, 2)
The derivative f′(x) = (x² − 1) / x² is 0 on the interval when x = 1 . To the left of that point the derivative is negative and to the right it is positive, so the function decreases and then increases, and it has a minimum point there: f(1) = 2/1 = 2 . The minimum point is (1, 2) .
Even and odd functions
For example: Here is the function f(x) = (x² + 6) / (x² − 2) . Is it even, odd, or neither?
- Odd
- Neither even nor odd
- Even
Answer and solution
The answer: Even
Put −x in place of x: f(−x) = (x² + 6) / (x² − 2) . That is exactly the given function, so it is even: its graph is symmetric about the vertical axis.
Finding a parameter
For example: The graph of the function f(x) = (x + a) / (x − 3) passes through the point (2, −3) . Find a .
Answer and solution
The answer: 1
Put in the point: −3 = (2 + a) / (2 − 3) . Multiply by the denominator: −3 × (−1) = 2 + a , so a = 1 .
Not on the site yet
Parts of this topic that are not built yet: Sketching the graph · Points where the numerator and the denominator are both 0 (not in the programme) · The second derivative (not in the programme).
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).