Find the domain of the function f(x)=x2−5x+4x2+3x+5
Get the complete, step-by-step math solution for: "Find the domain of the function f(x)= {x^{2}+3 x+5}{x^{2}-5 x+4}". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 x2−5x+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.