Answer:
See explanation below for further details.
Step-by-step explanation:
A rational consist of two real numbers such that:
[tex]\frac{a}{b} =c[/tex]
If c is a polynomial with a certain grade, then, both the numerator and the denominator must be also polynomials and the grade of the numerator must be greater than denominator.
If c is linear function, that is, a first order polynomial, then a must be a (n+1)-th polynomial and b must be a n-th polynomial.
Example:
If [tex]a = x^{2}[/tex] and [tex]b = x[/tex], then:
[tex]c = \frac{x^{2}}{x}[/tex]
[tex]c = x[/tex]
If c is a quadratic function, that is, a second order polynomial, then a must be a (n+1)-th polynomial and b must be a n-th polynomial.
Example
If [tex]a = 3\cdot x^{3}[/tex] and [tex]b = x[/tex], then:
[tex]c = \frac{3\cdot x^{3}}{x}[/tex]
[tex]c = 3\cdot x^{2}[/tex]
But if c is an exponential, both the numerator and the denominator must be therefore exponential function and grade of each exponential function must different to the other.
Example
If [tex]a = 10^{2x}[/tex] and [tex]b = 3\cdot 10^x[/tex], then:
[tex]c = \frac{10^{2\cdot x}}{3\cdot 10^{x}}[/tex]
[tex]c = \frac{1}{3}\cdot 10^{2\cdot x -x}[/tex]
[tex]c = \frac{1}{3}\cdot 10^x[/tex]
Otherwise, c would be equal to a constant function, that is, a polynomial with a grade 0.
If [tex]a = 5\cdot e^{x}[/tex] and [tex]b = -3\cdot e^{x}[/tex], then:
Example
[tex]c = \frac{5\cdot e^{x}}{-3\cdot e^{x}}[/tex]
[tex]c = -\frac{5}{3}[/tex]
It is worth to add that exponential functions can be a linear combination of single exponential function, similar to polynomials.
Example
[tex]5\cdot a^2\cdot x -9[/tex]
