Find the domain of the following [tex]f(x)=\frac{x+5}{x^2+1 x+2}[/tex]
A. [tex]x \neq-5[/tex]
B. All real numbers
C. [tex]x \neq 2,-1,-5[/tex]
D. [tex]x \neq 2,-1[/tex]
Answer (1)
The domain of a rational function is all real numbers except where the denominator is zero.
Set the denominator x 2 + x + 2 equal to zero and solve for x .
Calculate the discriminant b 2 − 4 a c to determine if there are real roots.
Since the discriminant is negative, there are no real roots, and the domain is all real numbers. All real numbers
Explanation
Understanding the Domain of a Rational Function We are given the function f ( x ) = x 2 + x + 2 x + 5 and asked to find its domain. The domain of a rational function is all real numbers except for values of x that make the denominator equal to zero. So, we need to find the values of x for which x 2 + x + 2 = 0 .
Applying the Quadratic Formula To find the roots of the quadratic equation x 2 + x + 2 = 0 , we can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c In this case, a = 1 , b = 1 , and c = 2 .
Calculating the Discriminant Let's calculate the discriminant, which is the part under the square root: b 2 − 4 a c = 1 2 − 4 ( 1 ) ( 2 ) = 1 − 8 = − 7
Analyzing the Discriminant Since the discriminant is negative ( − 7 < 0 ), the quadratic equation x 2 + x + 2 = 0 has no real roots. This means that the denominator x 2 + x + 2 is never equal to zero for any real number x .
Determining the Domain Therefore, the domain of the function f ( x ) = x 2 + x + 2 x + 5 is all real numbers.
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, when modeling the height of a projectile over time with a rational function, knowing the domain helps us determine the valid time intervals for which the model is meaningful. If the denominator of the function becomes zero at a certain time, the model is no longer valid at that point, indicating a limitation of the model or a physical constraint.
