Find the domain of the rational function.
[tex]f(x)=\frac{-3 x^2}{x^2+10 x-24}[/tex]
A. [tex]\langle x \mid x=-24,1\rangle[/tex]
B. [tex]\langle x \mid x=-12,2\rangle[/tex]
C. [tex]\langle x \mid x=12,-2\rangle[/tex]
D. [tex]\langle x \mid x=12,2\rangle[/tex]
Answer (1)
Find the values of x that make the denominator x 2 + 10 x − 24 equal to zero.
Factor the quadratic equation x 2 + 10 x − 24 = 0 into ( x + 12 ) ( x − 2 ) = 0 .
Solve for x to find x = − 12 and x = 2 .
Exclude these values from the domain, so the domain is all real numbers except x = − 12 and x = 2 , which can be written as { x ∣ x = − 12 , x = 2 } .
Explanation
Understanding the Problem We are given the rational function f ( x ) = x 2 + 10 x − 24 − 3 x 2 and asked to find its domain. The domain of a rational function consists of all real numbers except for the values of x that make the denominator equal to zero. Therefore, we need to find the values of x for which x 2 + 10 x − 24 = 0 .
Setting up the Equation To find the values of x that make the denominator zero, we need to solve the quadratic equation x 2 + 10 x − 24 = 0 . We can solve this equation by factoring, using the quadratic formula, or completing the square. Let's try factoring. We are looking for two numbers that multiply to -24 and add to 10. These numbers are 12 and -2.
Solving the Equation So, we can factor the quadratic equation as ( x + 12 ) ( x − 2 ) = 0 . This means that either x + 12 = 0 or x − 2 = 0 . Solving for x in each case, we get x = − 12 or x = 2 .
Finding the Domain Therefore, the denominator is zero when x = − 12 or x = 2 . These are the values that must be excluded from the domain of the function. The domain of f ( x ) is all real numbers except x = − 12 and x = 2 .
Final Answer In set notation, the domain is { x ∣ x = − 12 , x = 2 } .
Examples
Rational functions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can describe the relationship between distance, rate, and time when the rate is not constant. In economics, they can model cost-benefit ratios or supply-demand relationships. Understanding the domain of a rational function ensures that the model is valid and doesn't produce undefined or nonsensical results. For instance, if we are modeling the average cost of producing items, we need to make sure that the number of items produced does not make the denominator zero, as this would lead to an undefined average cost.
