Consider the following functions and their graphs:

c) Let's start from the function 11)(  xxh

Consider 11  xy and let's do some calculations:

 22 )1()1(11 xyxy  1)1( 2 xy 2)1(1  yx. This is the canonical equation of a parabola with vertex at point (1,1), concavity facing right and with axis of symmetry the line y = -1. Observe that, 101 ≤ x and that the square root 1+x can be calculated. Observe also that, 10101111  yyxyxy. Therefore, the function 11)(  xxh is such that -∞ < x ≤ 1 and -∞ < y < 1. Its graph is the branch of the parabola that is below the axis of symmetry.

The function 11)(  xxh is a parabola with vertex at point (1,1), concavity facing right and with axis of symmetry the line y = -1.

The domain of the function 11)(  xxh is -∞ < x ≤ 1 and the range is -∞ < y < 1.

The graph of the function 11)(  xxh is the branch of the parabola that is below the axis of symmetry.