Determine the left-hand behavior and the right-hand behaior of the graph of -3/4(x^3+2x^2-3x+1)?

Answer 1

Did you mean -(3/4) * (x^3+2x^2-3x+1) ?

Let f(x) = -(3/4) * (x^3+2x^2-3x+1). Equivalently, f(x) = -(3/4)x^3 - (3/2)x^2 + (9/4)x - 3/4. Consider an arbitrary real valued x say x = c. We need to determine if the left-hand behavior and the right-hand behavior of the function f at x = c are the same or not.

Since f is a polynomial function (of degree 3) on x, it follows that this is continuous on the set of real numbers. Thus, by applying the limit on both sides (left and right) of x, we will obtain the same value for both sides.

That is, the left-hand Limit will be as follows: as x approaches to the left of c, we make f(x) approaches f(c) = -(3/4)c^3 - (3/2)c^2 + (9/4)c - 3/4. Similarly, the right-hand limit will be: as x approaches to the right c, we also get f(x) approaches to f(c).

Therefore, both the left and right-hand limit yields to f(c); so as x approaches c in any direction, we always obtain f(c). Hence, both left and right-hand behavior of f are the same for any choice of a real valued x.

Answer 2

Right handed

Answer 3

Left behavior, goes to positive infinity

Right behavior, goes to negative infinity