Step 1

a. Function: \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\)

Step 2 For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: \(\displaystyle{x}^{{{2}}}-{1}={0}\)

Solve for x: \(\displaystyle{x}^{{{2}}}={1}\)

Apply square root on both sides of the equation: \(\displaystyle{x}=\sqrt{{{1}}}=\pm{1}\)

Step 3

The equation of vertical asymptote also gives information about the domain of this function, therefore the domain of \(\displaystyle{f{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,-{1}\right)}\cup{\left(-{1},{1}\right)}\cup{\left({1},\infty\right)}\)

Step 4

SInce the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is \(\displaystyle{y}={0}\)

Step 5

The range of this function \(\displaystyle{f{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,\infty\right)}.\) Step 6

Graph of \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\)

Step 7

b. Function: \(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\)

Step 8

For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: \(\displaystyle{x}^{{{2}}}+{3}{x}+{2}={0}\)

Factor using ac method: \(\displaystyle{x}^{{{2}}}+{x}+{2}{x}+{2}={0}\)

Factor: \(\displaystyle{\left({x}+{2}\right)}{\left({x}+{1}\right)}={0}\)

Solve for x: \(\displaystyle{x}=-{2}\)

And:\(\displaystyle{x}=-{1}\)

The equation of vertical asymptotes also gives information about the domain of this function, therefore the domain of \(\displaystyle{g{{\left({x}\right)}}}\ {i}{s}\ {\left(-\infty,-{2}\right)}\cup{\left(-{2},-{1}\right)}\cup{\left(-{1},\infty\right)}.\)

Step 9

Since the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is \(\displaystyle{y}={0}\)

Step 10

The range of this function \(g(x)\ is\ (-\infty, 0) \cup (13,928, \infty)\) (from graph)

Step 11

Graph of \(\displaystyle{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\)

Step 12

c.Function: \(\displaystyle{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}\)

Step 13

For vertical asymptote, equate the expression in the denominator with 0 and solve for x, therefore: \(\displaystyle{x}^{{{2}}}-{x}-{12}={0}\)

Factor using ac method: \(\displaystyle{x}^{{{2}}}-{4}{x}+{3}{x}-{12}={0}\)

Factor: \(\displaystyle{\left({x}-{4}\right)}{\left({x}+{3}\right)}={0}\)

Solve for x: \(\displaystyle{x}=-{3}\)

And: \(\displaystyle{x}={4}\)

Step 14

The equation of vertical asymptote also gives information about the domain of this function, therefore the domain of \(\displaystyle{h}{\left({x}\right)}\ {i}{s}\ {\left(-\infty,-{3}\right)}\cup\ {\left(-{3},{4}\right)}\cup{\left({4},\infty\right)}.\)

Step 15

SInce the degree of the expression in the denominator is 2, greater than the degree of the expression in the numerator, horizontal asymptote of this function is \(\displaystyle{y}={0}\).

Step 16

The range of this function \(h(x)\ is\ (-\infty, -0.398) \cup (-0.398, \infty)\) (from graph).

Step 17

\(\displaystyle{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}\)