Answer: The correct option is 4. The domain of f(x) is all real numbers except 2 and 4.
Explanation:
The given function is,
[tex]f(x)=\frac{x+1}{x^2-6x+8}[/tex]
Use factoring by grouping method to factorize the domain.
[tex]f(x)=\frac{x+1}{x^2-4x-2x+8}[/tex]
[tex]f(x)=\frac{x+1}{x(x-4)-2(x-4)}[/tex]
[tex]f(x)=\frac{x+1}{(x-4)(x-2)}[/tex]
The domain of a rational function is the intersection of domain of numerator or denominator expect those value for which the denominator function is 0.
The domain of numerator function (x+1) is all real number. The domain of denominator function (x-4)(x-2) is all real number.
The value of denominator equal to 0 when the value of x is 2 or 4. So the function f(x) is not defined for x=2 and x=4.
Therefore the domain of f(x) is all real numbers except 2 and 4. The fourth option is correct.
The domain of the given function, f(x)=x+1/x^2-6x+8 is; all real numbers except 2 and 4.
The domain of a function includes all values of the independent variable, usually x which renders a solution to the function.
In cases as described in the question;
The function f(x) is a quotient with its denominator as; x²-6x+8.
Hence, the zeros of the expression; x²-6x+8 will be exempted from the domain of f(x) since these zeros render the function undefined.
By factorisation;
Hence, the zeros are; x= 4 and x= 2
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