Question

1. Identify whether the functions shown are rational:

a. f(x) = x / x^2 + 1
b. f(x) = √x / x^2 + 1
c. f(x) = x / x^0.4 + 1
d. f(x) = (x / x^2 + 1)^2
e. f(x) = √2x / ex^2 + √3

Answer 1

a) Rational function

b) Not a rational function

c) Not a rational function

d) Rational function

e) Not a rational function

Explanation:

A rational function is of the form;

`f(x)=(p(x))/(q(x)) ,\:q(x)\\e 0` where
`p(x)` and
`q(x)` are polynomial functions.

For option a)
`f(x)=(x)/(x^2+1)`, we have both the numerator
`p(x)=x` and the denominator
`q(x)=x^2+1` are polynomial functions, hence
`f(x)=(x)/(x^2+1)` is a rational function.

For option b)
`f(x)=(√(x))/(x^2+1)`,
`p(x)=√(x)` is not a polynomial function hence
`f(x)=(√(x))/(x^2+1)` is not a rational function.

For option c) we have
`f(x)=(x)/(x^(0.4)+1)`,
`q(x)=x^(0.4)+1` is not a polynomial function hence
`f(x)=(x)/(x^(0.4)+1)` is not a rational function.

For option d)
`f(x)=((x)/(x^2+1))^2 =(x^2)/(x^4+2x^2+1)` both
`p(x)=x^2` and
`q(x)=x^4+2x^2+1` are polynomials hence
`f(x)=((x)/(x^2+1))^2` is a rational function.

For option e)
`f(x)=(√(2x) )/(e^(x^2)+√(3) )` both the numerator and the denominator are not polynomials hence
`f(x)=(√(2x) )/(e^(x^2)+√(3) )` is not a rational function.