Question

Show that the function given by \( f(x) = 12x - 3 \) is increasing on \( R \).

Hint:

For linear functions, the derivative is constant and if it is positive, the function is increasing for all real values of \( x \).

Answer 1

Step 1: Understand the function.
The function is \( f(x) = 12x - 3 \), which is a linear function. For a function to be increasing, its derivative must be positive.

Step 2: Differentiate the function.
The derivative of \( f(x) \) with respect to \( x \) is: \[ f'(x) = \frac{d}{dx} (12x - 3) = 12. \]

Step 3: Analyze the derivative.
Since \( f'(x) = 12 \) is positive for all values of \( x \), the function is increasing for all \( x \in R \).

Step 4: Conclusion.
Thus, the function \( f(x) = 12x - 3 \) is increasing on \( R \).