Rational functions and link with direct substitution property

[alingy1]

Homework Statement

Hello,

I know the direct substitution property in calculus. But, the definition of a rational function still confuses me.

For example, are these rational functions:

y=(x^2+2x+1)/(x+1)

y=((x^2+2)^(1/2))/(x+1)

The denominator of the first one could cancel. So, is there still a ratio? Is it a rational function?

The second one has a root in the numerator. The exponents of are not integers. Is it still a rational function?

Please help me. I do not understand the definition.

 

[UltrafastPED]

Rational expression: ratio of two polynomials; since 1 is a polynomial of order zero, y=x+1 is also a rational expression, as is y=1/x ... as long as you exclude zeros in the denominator.

A rational function is just a function which can be written as a rational expression. Your second example is not a rational expression, so y is not a rational function.

 

[alingy1]

Thank you for the fast reply.
But, what do you mean by rational expression? I do not quite get it. What about if there was an absolute value in the function?

EDIT: I understand why the second one is not a rational expression! Thank you. I still have a doubt about absolute values.

 

[UltrafastPED]

Rational expression: from ratio of expressions - in this case the individual expressions are always polynomials, so a rational expression is simply the ratio of two polynomials.

Your text may extend this to include absolute value of a ratio of polynomials, but that is not what is usually meant. But you always need to look at (and understand, and remember) the definition which is provided.