Consider the following function:
[tex]f(x)=\frac{2 x}{3 x^2-3}[/tex]
What is the domain of the function?
Answer (1)
Find values where the denominator equals zero: 3 x 2 − 3 = 0 .
Factor the equation: 3 ( x 2 − 1 ) = 0 ⟹ ( x − 1 ) ( x + 1 ) = 0 .
Solve for x : x = 1 and x = − 1 .
State the domain: All real numbers except x = − 1 and x = 1 , which is ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 3 x 2 − 3 2 x and we need to find its domain. The domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero.
Setting the Denominator to Zero To find the values of x that make the denominator zero, we set the denominator equal to zero and solve for x : 3 x 2 − 3 = 0
Factoring We can factor out a 3 from the equation: 3 ( x 2 − 1 ) = 0
Simplifying Divide both sides by 3: x 2 − 1 = 0
Factoring Difference of Squares Now, we can factor the quadratic expression as a difference of squares: ( x − 1 ) ( x + 1 ) = 0
Solving for x Setting each factor equal to zero, we get: x − 1 = 0 ⇒ x = 1 x + 1 = 0 ⇒ x = − 1
Determining the Domain Thus, the denominator is zero when x = 1 or x = − 1 . Therefore, the domain of the function is all real numbers except x = 1 and x = − 1 . In interval notation, the domain is ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ ) .
Final Answer The domain of the function f ( x ) = 3 x 2 − 3 2 x is all real numbers except x = − 1 and x = 1 .
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, when designing a bridge, engineers need to ensure that the load on the bridge (represented by a function) doesn't exceed certain limits to prevent collapse. The domain of this function would represent the range of loads the bridge can safely handle. Similarly, in economics, the demand function for a product has a domain that represents the range of prices for which there is a demand. Knowing the domain helps businesses make informed decisions about pricing and production.
