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.
A possible transformation of function f in the form of g(x) = f(x) + k can change the vertical position of the graph of f by adding or subtracting a constant k.
To determine whether there is a transformation that will result in a graph of y = g(x) with no x-intercepts, we need to consider the x-intercepts of f.
If f does not have any x-intercepts, then it is already true that its graph will have no x-intercepts. Adding or subtracting a constant k will not change the x-intercepts of f, as adding or subtracting a constant only shifts the graph vertically.
However, if f does have x-intercepts, then there may or may not be a transformation that will shift the graph of f vertically to eliminate all x-intercepts.
If the x-intercepts of f occur at points where f(x) + k is still negative (if k is positive) or positive (if k is negative), then there is no transformation that will eliminate all x-intercepts.
For example, consider the function f(x) = x^2 - 4. The graph of f has x-intercepts at x = -2 and x = 2. If we add a positive constant k to f(x), such as g(x) = x^2 - 4 + 3, the resulting graph will still have x-intercepts at x = -2 and x = 2. If we subtract a negative constant k from f(x), such as h(x) = x^2 - 4 - 3, the resulting graph will have x-intercepts at x = -\sqrt{7} and x = \sqrt{7}, which are not the same as the x-intercepts of f(x).
Therefore, the possibility of a transformation of f in the form of g(x) = f(x) + k that will result in no x-intercepts on the graph of y = g(x) depends on the specific x-intercepts of f and the value of k.