Question

# For the following exercises, find the domain of the rational functions. f(x)=\frac{x^{2}+4x-3}{x^{4}-5x^{2}+4}

Rational functions
For the following exercises, find the domain of the rational functions.
$$\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}+{4}{x}-{3}}}{{{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}}}}$$

2021-05-10
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.