Question

f is a quadratic function whose graph is a parabola opening upward and has a vertex on the x-axis. The graph of the new function g defined by g(x) = 2 - f(x - 5) has a range defined by the interval

Answer 1

If the graph of f is a parabola with vertex on x axis and opening up, then the range of f is given by the interval

$(0,+\infty )$
The graph of f(x - 5) is that of f(x) shifted 5 units to the right and therefore no change to the range. However since the graph of - f(x - 5) is that of f(x - 5) reflected on the x axis then the range of - f(x - 5) is given by
$(-\infty ,0)$
The graph g(x) = 2 - f(x - 5) is that of - f(x - 5) shifted up by 2 units; hence the range of g(x) is given by
$(-\infty ,2)$