Solved: "Show that f(x)=x2+3x−1 and g(x)=3x3−9x+x−2 are rational functions..."

Anish Buchanan

Answered question

2021-09-03

Show that

f (x) = x^{2} + 3 x^{- 1}

and

g (x) = 3 x^{3} - 9 x + x^{- 2}

are rational functions that is, quotients of polynomials.

Nola Robson

Skilled2021-09-04Added 94 answers

To show:
That the functions are rational function:
Given:
The functions are

f (x) = x^{2} + 3 x^{- 1}

and

g (x) = 3 x^{3} - 9 x + x^{- 2}

Concept used:
The function h(x) is rational function if h(x) is represented as

\frac{p (x)}{q (x)}

Here, p(x) and q(x) are polynomial function.
Verification:
Check first function

f (x) = x^{2} + 3 x^{- 1}

f (x) = x^{2} + 3 \times \frac{1}{x}

= \frac{x^{3} + 3}{x}

Therefore, the function

f (x) = x^{2} + 3 x^{- 1}

can be represented as

\frac{p (x)}{q (x)}

, then the function

f (x) = x^{2} + 3 x^{- 1}

is rational function.
Check first function

g (x) = 3 x^{3} - 9 x + x^{- 2}

g (x) = 3 x^{3} - 9 x + x^{- 2}

= 3 x^{3} - 9 x + \frac{1}{x^{2}}

= \frac{3 x^{5} - 9 x^{3} + 1}{x^{2}}

Therefore, the function

g (x) = 3 x^{3} - 9 x + x^{- 2}

can be represented

\frac{p (x)}{q (x)}

, then the function

g (x) = 3 x^{3} - 9 x + x^{- 2}

is ratopnal function.

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