Q. what are the domain and range of (f(x) = 2(3x))?

Problem: Find the domain and range of the function \( f(x) = 2(3x) \).

1. Step 1: Simplify the function

   The first step is to simplify the expression by performing the multiplication. Since \(2\times 3 = 6\), we can rewrite the function as:

   \[ f(x) = 6x \]

   This is a linear function in the form \( f(x) = mx + b \), where the slope \( m = 6 \) and the y-intercept \( b = 0 \).

2. Step 2: Determine the domain

   The domain of a function is the set of all possible input values for \( x \) that result in a defined real number. For the linear function \( f(x) = 6x \), there are no mathematical restrictions such as:

   • Division by zero
   • Square roots of negative numbers
   • Logarithms of non-positive numbers

   Since any real number can be multiplied by 6, the domain is the set of all real numbers.

   Domain: \( (-\infty, \infty) \) or all real numbers.

3. Step 3: Determine the range

   The range is the set of all possible output values for \( f(x) \). For a linear function with a non-zero slope, the output will continue to increase as \( x \) increases and decrease as \( x \) decreases. There is no maximum or minimum value. Because the graph is a straight line that extends infinitely in both directions, every real number is a possible output.

   Range: \( (-\infty, \infty) \) or all real numbers.

4. Final Answer:

   Domain: \( (-\infty, \infty) \)

   Range: \( (-\infty, \infty) \)