Function:

\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}+{4}{x}-{3}}}{{{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}}}}\)

2. For a function that is defined by a fraction, the function is undefined when its denominator is equal to 0, therefore equate the denominator with 0 to evaluate the value of « for which the function will be undefined:

\(\displaystyle{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}={0}\)

Factor using ac method:

\(\displaystyle{x}^{{{4}}}-{x}^{{{2}}}-{4}{x}^{{{2}}}+{4}={0}\)

Write as factors: \(\displaystyle{\left({x}^{{{4}}}-{1}\right)}{\left({x}^{{{2}}}-{4}\right)}={0}\)

Each of these factors can further be factored using difference of squares method, therefore:

(x-1)(x+1)(x-2)(x+2)=0

Now equate each factor with 0 and solve for the variable, therefore:

x=-2

x=-1

x=1

x=2

Therefore, it can be concluded that the domain of the function are all real numbers except x=-2,x=-1,x=1 and x=2.

\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}+{4}{x}-{3}}}{{{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}}}}\)

2. For a function that is defined by a fraction, the function is undefined when its denominator is equal to 0, therefore equate the denominator with 0 to evaluate the value of « for which the function will be undefined:

\(\displaystyle{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}={0}\)

Factor using ac method:

\(\displaystyle{x}^{{{4}}}-{x}^{{{2}}}-{4}{x}^{{{2}}}+{4}={0}\)

Write as factors: \(\displaystyle{\left({x}^{{{4}}}-{1}\right)}{\left({x}^{{{2}}}-{4}\right)}={0}\)

Each of these factors can further be factored using difference of squares method, therefore:

(x-1)(x+1)(x-2)(x+2)=0

Now equate each factor with 0 and solve for the variable, therefore:

x=-2

x=-1

x=1

x=2

Therefore, it can be concluded that the domain of the function are all real numbers except x=-2,x=-1,x=1 and x=2.