I was not really prepared to answer, but given that you asked in an A2A, and given that the answers so far did not understand the question, I will try.
First, the definition is needed. A function f is rational if and only if it is the quotient of two polynomials… that is, expressible in the form
f(x) = P(x) / Q(x) where P and Q are polynomials.
In particular, if P and Q have degree 1, then f(x) = (ax + b) / (cx + d)
where a, b, c, and d are constants. My first examples will both use this.
A specific example would be f(x) = x / (x + 1).
Customarily, the ratio should be in lowest terms, such as
(x + 2) / x but NOT (3x + 6) / 3x (because of common factor 3).
For example, an inverse relationship, mathematically y = 1 / x, fits this, as one has a = 1, and b, c, d each zero. A real-life example of this is the ideal gas law:
P = k / V where k is a constant, P = pressure, V = volume of gas.
Second example. For purposes of illustration, imagine you and your son are immortal. If you are 30 years older than your son (because he was born when you were age 30), in year x of his life, the ratio of your age to his is
(x + 30 / x, or the ratio of his to yours is x / (x + 30).
In either case, as time goes on, both x and x + 30 (your ages) increase, but either ratio I just gave approaches 1, even though you are always 30 years apart in age. To some people, this feels paradoxical, because if the ratio were exactly 1, the ages would have to be identical! (Of course, the ratio only approaches 1 but never reaches 1.)
Interestingly, functions of the form f(x) = (ax + b) / (cx + d) generally are invertible, with the inverse function also rational.
There is no need for the polynomials to be linear, as in next example:
An object fired straight up from the ground at time t = 0 with instantaneous velocity 100 ft/sec will have height above earth (i.e., position) given by
s = -16 t^2 + 100 t
and, in this case, for the seconds projectile remains in the air,
the ratio (-16 t^2 + 100t) / t is a rational function that gives the average velocity from time t = 0 to time t.
Answers & Comments
Answer:
Step-by-step explanation:
I was not really prepared to answer, but given that you asked in an A2A, and given that the answers so far did not understand the question, I will try.
First, the definition is needed. A function f is rational if and only if it is the quotient of two polynomials… that is, expressible in the form
f(x) = P(x) / Q(x) where P and Q are polynomials.
In particular, if P and Q have degree 1, then f(x) = (ax + b) / (cx + d)
where a, b, c, and d are constants. My first examples will both use this.
A specific example would be f(x) = x / (x + 1).
Customarily, the ratio should be in lowest terms, such as
(x + 2) / x but NOT (3x + 6) / 3x (because of common factor 3).
For example, an inverse relationship, mathematically y = 1 / x, fits this, as one has a = 1, and b, c, d each zero. A real-life example of this is the ideal gas law:
P = k / V where k is a constant, P = pressure, V = volume of gas.
Second example. For purposes of illustration, imagine you and your son are immortal. If you are 30 years older than your son (because he was born when you were age 30), in year x of his life, the ratio of your age to his is
(x + 30 / x, or the ratio of his to yours is x / (x + 30).
In either case, as time goes on, both x and x + 30 (your ages) increase, but either ratio I just gave approaches 1, even though you are always 30 years apart in age. To some people, this feels paradoxical, because if the ratio were exactly 1, the ages would have to be identical! (Of course, the ratio only approaches 1 but never reaches 1.)
Interestingly, functions of the form f(x) = (ax + b) / (cx + d) generally are invertible, with the inverse function also rational.
There is no need for the polynomials to be linear, as in next example:
An object fired straight up from the ground at time t = 0 with instantaneous velocity 100 ft/sec will have height above earth (i.e., position) given by
s = -16 t^2 + 100 t
and, in this case, for the seconds projectile remains in the air,
the ratio (-16 t^2 + 100t) / t is a rational function that gives the average velocity from time t = 0 to time t.