Every polynomial p(x) can be represented as \(\displaystyle{p}\frac{{{x}}}{{1}}\). As y=1 is trivially a polynomial we conclude that the statement is TRUE.

asked 2021-05-22

Describe the key characteristics of the graphs of rational functions of the form \(\displaystyle{f{{\left({x}\right)}}}=\frac{{{a}{x}+{b}}}{{{c}{x}+{d}}}\). Explain how you can determine these characteristics using the equations of the functions. In what ways are the graphs of all the functions in this family alike? In what ways are they different? Use examples in your comparison.

asked 2021-05-17

asked 2021-05-03

Consider the following rational functions:
\(\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}-{1}}}{{{x}^{{2}}-{x}-{2}}}\).

\(\displaystyle{s}{\left({x}\right)}=\frac{{{x}^{{3}}+{27}}}{{{x}^{{2}}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{x}^{{3}}-{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}-{6}}}{{{x}^{{2}}-{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{x}^{{3}}+{6}{x}^{{2}}+{9}{x}}}{{{x}+{3}}}\)

What are the asymptotes of the function r(x)?

\(\displaystyle{s}{\left({x}\right)}=\frac{{{x}^{{3}}+{27}}}{{{x}^{{2}}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{x}^{{3}}-{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}-{6}}}{{{x}^{{2}}-{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{x}^{{3}}+{6}{x}^{{2}}+{9}{x}}}{{{x}+{3}}}\)

What are the asymptotes of the function r(x)?

asked 2021-06-07

\(\displaystyle{s}{\left({x}\right)}=\frac{{{x}^{{3}}+{27}}}{{{x}^{{2}}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{x}^{{3}}-{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}-{6}}}{{{x}^{{2}}-{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{x}^{{3}}+{6}{x}^{{2}}+{9}{x}}}{{{x}+{3}}}\)

What are the asymptotes of the function r(x)?

asked 2021-05-25

Choose the correct term from the list above to complete each sentence. Power, radical, polynomial. and rational functions are examples of _____.

asked 2021-05-14

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions f/g, with the degree off ≤≤ the degree ofg (including f = 0).

asked 2021-06-23

This ideal gas law states that the volume of
an enclosed gas at a fixed temperature varies inversely as the
pressure.

Which power functions are also rational functions?

Which power functions are also rational functions?

asked 2021-05-01

Consider the following rational functions:
\(\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}−{1}}}{{{\left({x}^{{2}}\right)}−{x}−{2}}}\)

\(\displaystyle{s}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{27}}}{{{\left({x}^{{2}}\right)}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{\left({x}^{{2}}\right)}−{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}}}{{{x}+{3}}}\)

Which of these rational functions has a horizontal asymptote?

\(\displaystyle{s}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{27}}}{{{\left({x}^{{2}}\right)}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{\left({x}^{{2}}\right)}−{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}}}{{{x}+{3}}}\)

Which of these rational functions has a horizontal asymptote?

asked 2021-05-26

Consider the following rational functions:
\(\displaystyle{r}{\left({x}\right)}=\frac{{{2}{x}−{1}}}{{{x}^{{2}}}}−{x}−{2}{s}{\left({x}\right)}=\frac{{{x}^{{3}}+{27}}}{{{x}^{{2}}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{x}^{{2}}−{25}}}\)

\(\displaystyle{w}{\left({x}\right)}={\left({\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}\right)}\frac{{)}}{{{x}+{3}}}\)

Which of these functions has no vertical asymptote?

\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{x}^{{2}}−{25}}}\)

\(\displaystyle{w}{\left({x}\right)}={\left({\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}\right)}\frac{{)}}{{{x}+{3}}}\)

Which of these functions has no vertical asymptote?

asked 2021-06-05

\(\displaystyle{s}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{27}}}{{{\left({x}^{{2}}\right)}+{4}}}\)

\(\displaystyle{t}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}−{9}{x}}}{{{x}+{2}}}\)

\(\displaystyle{u}{\left({x}\right)}=\frac{{{\left({x}^{{2}}\right)}+{x}−{6}}}{{{\left({x}^{{2}}\right)}−{25}}}\)

\(\displaystyle{w}{\left({x}\right)}=\frac{{{\left({x}^{{3}}\right)}+{\left({6}{x}^{{2}}\right)}+{9}{x}}}{{{x}+{3}}}\)

Which of these rational functions has a horizontal asymptote?