The reciprocal of a function · Grade 11 · 4 units
In The reciprocal of a function (Grade 11 · 4 units) you practise 6 skills: The domain of the reciprocal · Asymptotes and the horizontal axis · Where it is positive or negative · Where the reciprocal goes · From a function to its reciprocal · Shifts, reflections and stretches. The reciprocal of a function is one divided by the function. It is not defined where the given function is 0, and its graph has a vertical asymptote there. It is never 0 itself, because a fraction whose numerator is 1 is never 0.
The reciprocal always has the same sign as the given function. Where the given function increases, the reciprocal decreases, and where it decreases, the reciprocal increases. A positive minimum of the given function is a maximum of the reciprocal. When the given function grows without bound in size, the reciprocal approaches 0, so the horizontal axis is its horizontal asymptote. The graph of one divided by x minus a number, plus another number, is the graph of one divided by x moved, and its asymptotes move with it. A reflection in the horizontal axis changes the sign of every value, a reflection in the vertical axis puts minus x in place of x, a vertical stretch multiplies every value by a number, and the absolute value reflects the parts below the horizontal axis.
Tip: Before any question, find where the given function is 0: that is exactly where the reciprocal has its vertical asymptotes.
What you learn
The domain of the reciprocal
For example: Here is the function f(x) = x + 5 , and the function g(x) = 1 / f(x) . For which value of x is g not defined?
Answer and solution
The answer: −5
The given function x + 5 is 0 when x = −5 . There g is not defined.
Asymptotes and the horizontal axis
For example: Here is the function f(x) = x + 5 , and the function g(x) = 1 / f(x) . Find the vertical asymptote of the graph of g .
Answer and solution
The answer: −5
The given function is 0 when x = −5 , and there g grows or falls without bound. The vertical asymptote is x = −5 .
Where it is positive or negative
For example: Here is the function f(x) = x + 5 , and the function g(x) = 1 / f(x) . For which values of x is g(x) > 0 ?
Answer and solution
The answer: x > −5
The reciprocal has the same sign as the given function, and the given function is positive when x > −5 .
| x < −5 | x > −5 |
|---|---|
| − | + |
Where the reciprocal goes
For example: Here is the function f(x) = −4x + 6 , and the function g(x) = 1 / f(x) . What happens to g as x → ∞ ?
- It grows without bound
- It falls without bound
- It approaches 1
- It approaches 0
Answer and solution
The answer: It approaches 0
Intuitively: as x grows without bound, the given function grows or falls without bound, so 1 divided by it gets very small in size: it approaches 0.
From a function to its reciprocal
For example: The quadratic function f has its minimum point at (−4, 4) , and g(x) = 1 / f(x) . At which value of x does g have a turning point?
Answer and solution
The answer: −4
The given function is positive everywhere, and it is smallest at x = −4 . There 1 divided by it is largest, so g has its maximum point there: g(−4) = 1 / 4 .
Shifts, reflections and stretches
For example: Here is the function g(x) = 1 / (x + 4) + 5 . Find the point where the two asymptotes of its graph meet.
Answer and solution
The answer: (−4, 5)
The denominator is 0 when x = −4 , and that is the vertical asymptote. Far from the origin the fraction gets closer to 0, so the horizontal asymptote is y = 5 . They meet at (−4, 5) .
Not on the site yet
Parts of this topic that are not built yet: Sketching the graph · Limits as a calculated answer.
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).