If the range of function y = f(x) is f(x) is grater then or equal to 11, then the range of the new function g(x) = f(x - 3) + 2 is y which is greater then or equal to k?

Question

If the range of function y = f(x) is f(x) is grater then or equal to 11, then the range of the new function g(x) = f(x - 3) + 2 is y which is greater then or equal to k?

Answer 1

If f(x) has a range where y≥11, then where is the range of f(x - 3) + 2?

The 3 in (x - 3) causes a horizontal shift of the graph, to the right by 3 units. All the y-values remain intact, that is the range is still y≥11.

But the 2 in f(x - 3) + 2 is "outside" the parentheses and causes changes to the y-values (an increase of 2). That is the graph of f is vertically shifted up 2. That means the lowest y-value is 11+2 = 13. So the range of the transformed function is y ≥ 13.

This graph shows a parabola that has a range y ≥ 11, i.e. it is an upward opening parabola with its vertex on the line y = 11. Then is shows functions where the parabola is shifted to the right 2 (no change in range, and another where there is a vertical shift up 2.

desmos.com/calculator/vut6eu9kaq