Explanation:
it says it in the headline : the cube root function.
1.
it goes through (0, 0) and (1, 1) and (-1, -1).
so,
f(x) = cubic root(x)
domain = (-infinity, +infinity)
range = (-infinity, +infinity)
x-intercept = 0
y-intercept = 0
2.
the function is y = cubic root(x + 2)
compared to
y = cubic root(x)
the domain is x >= -2
wrong !
this is not a square root. for the square root that domain would be correct (there are no real number square roots for negative values). but cubic roots are possible also for negative values. this function has the same domain and range as the basic y = cubic root(x) : (-infinity, +infinity).
the function is a vertical translation of g(x) = cubic root(x).
wrong !
vertical means up/down.
but it is a horizontal (left/right) translation of that g(x). by 2 units to the left, actually.
3.
for 1 <= x <= 4 we have already the average rate of change :
about 0.197
for 4 <= x <= 7 we get
(f(7) - f(4))/(7 - 4) ≈ (1.91 - 1.59)/3 ≈ 0.109
so, the interval [1, 4] has a greater average of change.
which is also clear when looking at the graph : the function increases stronger for the lower x-values than for the larger x-values.