Find the domain of the function $f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$

Step 1: Identify the restriction for rational functions

For a rational function, the denominator cannot be equal to zero. If the denominator were zero, the function would be undefined at that point. Therefore, we must find the values of $x$ that make the denominator zero and exclude them from the domain.

Step 2: Factor the quadratic denominator

To find the values of $x$ that make the denominator zero, we factor the quadratic expression $x^2 - 5x + 4$. We look for two numbers that multiply to 4 and add up to -5. These numbers are -1 and -4, so the factored form is $(x-1)(x-4)$.

Step 3: Solve for x

Setting each factor equal to zero gives us the values of $x$ that make the denominator zero. Thus, $x=1$ and $x=4$ are the values for which the function is undefined.

Step 4: State the domain

The domain of the function includes all real numbers except for the values that make the denominator zero. Therefore, the domain is all real numbers except 1 and 4.

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