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
ANSWERED
asked 2021-05-09
For the following exercises, find the domain of the rational functions.
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}^{{{2}}}+{4}{x}-{3}}}{{{x}^{{{4}}}-{5}{x}^{{{2}}}+{4}}}}\)

Answers (1)

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.
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