state whether there is a possible transformation of f in the form g(x) = f (x + k), where k is a constant, for which the graph of y = g(x) will have no x-intercepts. Justify your answer.
If f(x) has an x-intercept (a point where f(x) = 0), then there exists a value of x such that f(x + k) = 0 for some constant k, because shifting the graph horizontally by k units does not affect the y-intercepts. Therefore, there is no possible transformation of f in the form g(x) = f(x + k) that will eliminate x-intercepts.
Alternatively, we can use the fact that a horizontal shift of k units does not change the shape or direction of the graph of f, only its position. Thus, if f has an x-intercept, shifting it horizontally by any amount will not change the fact that it intersects the x-axis.