The domain of [tex]f(x)=x^2 \ln x[/tex] is
A. [tex](-\infty, \infty)[/tex]
B. [tex](1, \infty)[/tex]
C. [tex](0,1)[/tex]
D. [tex](0, \infty)[/tex].
Answer (2)
The domain of the function f ( x ) = x 2 ln x is determined by the domains of its components. Since x 2 is defined for all real numbers, and ln x is defined for positive x , the domain of f ( x ) is ( 0 , ∞ ) . Therefore, the correct answer is D.
;
The domain of x 2 is all real numbers, ( − ∞ , ∞ ) .
The domain of ln x is ( 0 , ∞ ) .
The domain of f ( x ) = x 2 ln x is the intersection of the domains of x 2 and ln x .
The domain of f ( x ) is ( 0 , ∞ ) .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = x 2 "." ln x . The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Analyzing the Components The function f ( x ) is a product of two functions: x 2 and ln x . Let's analyze the domain of each function separately.
Domain of x^2 The function x 2 is a polynomial, and polynomials are defined for all real numbers. So, the domain of x 2 is ( − ∞ , ∞ ) .
Domain of ln(x) The function ln x (natural logarithm of x) is only defined for positive values of x . So, the domain of ln x is ( 0 , ∞ ) .
Finding the Intersection Since f ( x ) is the product of x 2 and ln x , it is defined only where both x 2 and ln x are defined. Therefore, the domain of f ( x ) is the intersection of the domains of x 2 and ln x .
Determining the Domain of f(x) The intersection of ( − ∞ , ∞ ) and ( 0 , ∞ ) is ( 0 , ∞ ) . This means that f ( x ) is defined for all positive real numbers.
Final Answer Therefore, the domain of f ( x ) = x 2 ln x is ( 0 , ∞ ) . The correct answer is D.
Examples
Understanding the domain of functions is crucial in many real-world applications. For example, in physics, if f ( t ) represents the position of an object at time t , and f ( t ) = t 2 ln t , then the domain tells us for what times the position is defined. Since time cannot be negative or zero in this context (due to the logarithm), the domain ( 0 , ∞ ) indicates that we can only consider the object's position for positive times.
