Question

How are rational functions similar to linear, quadratic, or exponential functions? How are they different? When are these similarities or differences important when looking for intersections between rational functions and other types of functions? Be sure to cite examples as you explain your ideas.

Answer 1

See explanation below for further details.

Explanation:

A rational consist of two real numbers such that:

`(a)/(b) =c`

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
`a = x^(2)` and
`b = x`, then:

`c = (x^(2))/(x)`

`c = x`

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
`a = 3\cdot x^(3)` and
`b = x`, then:

`c = (3\cdot x^(3))/(x)`

`c = 3\cdot x^(2)`

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
`a = 10^(2x)` and
`b = 3\cdot 10^x`, then:

`c = (10^(2\cdot x))/(3\cdot 10^(x))`

`c = (1)/(3)\cdot 10^(2\cdot x -x)`

`c = (1)/(3)\cdot 10^x`

Otherwise, c would be equal to a constant function, that is, a polynomial with a grade 0.

If
`a = 5\cdot e^(x)` and
`b = -3\cdot e^(x)`, then:

Example

`c = (5\cdot e^(x))/(-3\cdot e^(x))`

`c = -(5)/(3)`

It is worth to add that exponential functions can be a linear combination of single exponential function, similar to polynomials.

Example

`5\cdot a^2\cdot x -9`