Given f : R → R defined by f(x) = x · e x and g : R → R defined by g(x) = √x² + 1.
(a) Find f ο g
(b) Find g ο f
(c) Find the range of g and state the range of g in correct set notation
(d) Let S = (−1, 1) = {x ∈ R | − 1 < x < 1}. Find g(S) and express g(S) in correct set notation.
(e) Let T = {√2}. Find g⁻¹ (T) and express g⁻¹ (T) in correct set notation.

  the distance between the points is about 9.2 units

Step-by-step explanation:

It is well you should not understand it. No question is asked.

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The answer choices suggest you are to find the distance between the two points. There is only one choice in a reasonable range: 9.2 units.

Each point is more than 2 units from any axis, so 2 units is clearly not the answer. The size of the graph is much less than 81 units, so clearly that is not the answer.

The difference of coordinates in the x-direction is 6; in the y-direction the difference is 7 units. The distance between the points will be more than the longest of these (7) and less than about 1.5 times that (10.5). Only one choice is in this range: 9.2 units.

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The Pythagorean theorem is used to calculate the distance between points. The distance is considered to be the hypotenuse of a right triangle with legs of lengths equal to the differences of coordinates. Here, that means the distance (d) is ...

  d² = 6² + 7² = 36 +49 = 85

  d = √85 ≈ 9.2 . . . . grid squares, or "units"