91Ó°ÊÓ

Given \(f'(x) = 2\), it's clear that the derivative is a constant function. A constant function has a horizontal line as its graph. So, plot a horizontal line on the coordinate plane with the equation \(y = 2\).

Since the derivative, \(f'(x)\), is always positive (equal to 2), it means that the original function \(f(x)\) is always increasing. Additionally, the derivative is constant, which means that \(f(x)\) has a constant rate of increase (i.e., its slope is constant). This information tells us that \(f(x)\) must be a linear function.

We know that \(f(x)\) is a linear function with a constant slope of 2, which means that its general equation is \(f(x) = 2x + C\), where \(C\) is an arbitrary constant that represents the intercept on the y-axis. Varying the value of C will give us different possible graphs of \(f(x)\). Let's sketch three such graphs: 1. Graph of \(f(x) = 2x\): Set \(C = 0\). Plot the line with equation \(y = 2x\). It will pass through the origin \((0,0)\) and have a slope of 2. 2. Graph of \(f(x) = 2x + 1\): Set \(C = 1\). Plot the line with equation \(y = 2x + 1\). It will pass through the point \((0,1)\) and have a slope of 2. 3. Graph of \(f(x) = 2x - 1\): Set \(C = -1\). Plot the line with equation \(y = 2x - 1\). It will pass through the point \((0,-1)\) and have a slope of 2. So, by looking at the graph of \(f'(x)\) and understanding the relationship between a function and its derivative, we have been able to sketch three different possible graphs of the original function \(f(x)\).