Question

What is the nature of function f(x) = 7x-4 on R?

• A. Increasing
• B. Decreasing
• C. Strictly Increasing
• D. Increasing and Decreasing

Answer 1

Answer: (C)

C

Answer 2

To determine the nature of the function $f(x) = 7x - 4$ on R, we can use the concept of derivatives.

1. Find the first derivative of the function: Given $f(x) = 7x - 4$. Differentiating $f(x)$ with respect to $x$: $f'(x) = \frac{d}{dx}(7x - 4)$ $f'(x) = 7 - 0$ $f'(x) = 7$

2. Analyze the sign of the derivative: The derivative $f'(x) = 7$. Since $f'(x) = 7 > 0$ for all $x \in R$, the function is strictly increasing on R.

3. Definition of increasing and strictly increasing functions:

   • A function $f(x)$ is increasing on an interval if for any $x_1 < x_2$ in that interval, $f(x_1) \le f(x_2)$.

   • A function $f(x)$ is strictly increasing on an interval if for any $x_1 < x_2$ in that interval, $f(x_1) < f(x_2)$.

   In our case, for any $x_1 < x_2$: $7x_1 < 7x_2$ $7x_1 - 4 < 7x_2 - 4$ $f(x_1) < f(x_2)$ This confirms that the function is strictly increasing.

Since $f'(x) = 7$ is always positive, the function $f(x) = 7x - 4$ is strictly increasing on the entire real line R. While it is also "increasing" (as strictly increasing implies increasing), "Strictly Increasing" is the most precise and accurate description.