Question

Which of the following is a rational function?
$A)\dfrac{1}{3}\sqrt {4{x^3} + 4x + 7}$
$B)\dfrac{{3{x^3} - 7x + 1}}{{x - 2}},x \ne 2$
$C)\dfrac{{3{x^5} + 5{x^3} + 2x + 7}}{{{x^{\dfrac{3}{2}}}}},x > 0$
$D)\dfrac{{\sqrt {1 + x} }}{{2 + 5x}},x \ne \dfrac{{ - 2}}{5}$

Answer 1

First we have to define what the terms we need to solve the problem are.
First, we need to know about the rational numbers, since natural numbers contained in whole numbers, whole numbers contained in integers, integers contained in rational and irrational numbers; the only difference between rational and irrational is both have decimal facts but rational will be in the form of $\dfrac{p}{q},q \ne 0$ an irrational will be in the form of non-rational numbers but it can also be written in fractions like$\sqrt 2 = 1.414$, and we need to find the rational functions from the given set of values.

Complete step-by-step solution:
Since rational numbers can be expressed as a fraction that contains numerator and denominator terms and that denominator will be never zero; if suppose it is zero then we clearly say that value is an infinite term. Now we are going to check the options A, B, C, D which are all rational functions using the definition.
First for an option $A)\dfrac{1}{3}\sqrt {4{x^3} + 4x + 7}$ that is an irrational number (irrational can be expressed as square root terms) hence it is wrong.
For option, $B)\dfrac{{3{x^3} - 7x + 1}}{{x - 2}},x \ne 2$ both numerator and denominators are rational numbers, and hence it is a rational function.
For option $C)\dfrac{{3{x^5} + 5{x^3} + 2x + 7}}{{{x^{\dfrac{3}{2}}}}},x > 0$ is an irrational number since ${x^{\dfrac{3}{2}}}$ the denominator is irrational (as they contain root terms) and finally for the option $D)\dfrac{{\sqrt {1 + x} }}{{2 + 5x}},x \ne \dfrac{{ - 2}}{5}$ is irrational too as it contains the root terms.
Hence the only option correct is $B)\dfrac{{3{x^3} - 7x + 1}}{{x - 2}},x \ne 2$ both terms are rational thus it is a rational function.

Note: Since the rational number is in the form of a function and that is called the rational function. Like $3{x^3} - 7x + 1$ is a rational number and also $x - 2$ is again a rational number hence it is a rational function. And the number which does not satisfy the condition of rational numbers and also in terms of decimal is known as irrational numbers.