Q. What are the Domain and Range of \( f(x) = 2(3x) \)?

Solution

1. Simplify the expression.

   The given function is \( f(x) = 2(3x) \). Multiply the constants to simplify:
   
   \( f(x) = 2 \cdot 3x = 6x \).

2. Determine the domain.

   The domain is the set of all input values \(x\) for which the formula makes sense. The simplified formula is \( f(x)=6x \). This is a polynomial (in fact, a linear function). Polynomials are defined for every real number because there are no denominators, square roots of negative numbers, logarithms, or other operations that restrict inputs. Therefore every real number can be used as an input.

   So the domain is the set of all real numbers: \( \mathbb{R} \), which in interval notation is \( (-\infty,\infty) \).

3. Determine the range.

   The range is the set of all output values \(y=f(x)\) that the function can produce. For \( f(x)=6x \), solve for \(x\) in terms of an arbitrary real output \(y\):
   
   Suppose \( y \) is a real number and \( y = 6x \). Solve for \( x \):
   
   \( x = \dfrac{y}{6} \).

   Because \( \dfrac{y}{6} \) is a real number for every real \(y\), for each real \(y\) there exists an \(x\) (namely \(x=y/6\)) such that \( f(x)=y \). Thus every real number appears as an output.

   Therefore the range is the set of all real numbers: \( \mathbb{R} \), which in interval notation is \( (-\infty,\infty) \).

Answer: Domain: \( \mathbb{R} \) (or \( (-\infty,\infty) \)). Range: \( \mathbb{R} \) (or \( (-\infty,\infty) \)).