To determine the window for which the visible portion of the graph looks like a cubic function or a square root function, we need to consider the shape and behavior of those functions.
a. A cubic function is a polynomial function with a degree of 3. It typically has two "hills" or "valleys" in its graph. To create a window that resembles a cubic function, we need to select a window that shows two hills or valleys.
b. A square root function is a function of the form ƒ(x) = √x, where the graph resembles the shape of the square root symbol (√). To create a window that resembles a square root function, we need to select a window that shows a curve that starts at the origin and increases without bound.
Since the given function is f(x) = 3x / (4 - x²), we can analyze the behavior of the graph to determine the appropriate window.
1. For a cubic function, we expect to see two hills or valleys. Looking at the given function, we can see that it is a rational function with a denominator of 4 - x². This indicates that the graph may have vertical asymptotes at x = 2 and x = -2. To capture the two hills or valleys, we should select a window that includes these vertical asymptotes.
2. For a square root function, we expect to see a curve that starts at the origin and increases without bound. Looking at the given function, we can see that the numerator is a linear function (3x) and the denominator is a quadratic function (4 - x²). This indicates that the graph may have a vertical asymptote at x = -2 and a horizontal asymptote at y = 0. To capture the curve starting at the origin and increasing without bound, we should select a window that includes the origin and extends towards the positive x-axis.
In summary:
a. To create a window that resembles a cubic function, we should select a window that includes the vertical asymptotes at x = 2 and x = -2.
b. To create a window that resembles a square root function, we should select a window that includes the origin and extends towards the positive x-axis.
Keep in mind that the specific window dimensions (x-range and y-range) will depend on the scale of the graph and the specific values of the function.