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21.06.2019 •
Mathematics
Lindsay is creating a bridge out of dried pasta and glue. in her plans, she sketches the graph to determine the lengths of the noodles, in centimeters, needed for the arch to increase the length of the bridge in 1 centimeter increments.which function did lindsay use to create her design? f(x) = (x – 5)2 + 8 f(x) = –(x + 5)2 + 8 f(x) = 0.5(x + 5)2 + 8 f(x) = –0.5(x – 5)2 + 8
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Ответ:
If
is the inverse of
, then
Given that
, we have
Solve for the inverse :
At this point, we have two possible solutions:
• If
, then
. Continuing with the equation, we have
• Otherwise, if
, then
, in which case
Both solutions are simultaneously incompatible. For instance, when x = 0 we have
using the first solution, and
using the second one. Choosing one over the other depends on how you restrict the domain.
(A) If we want to stick with the first solution, we would required that x ≥ 2. In other words, we would have
defined only when x ≥ 2; then
is defined only when
, so that ...
(B) ... the inverse is
(C) The domain of the inverse is the same as the range of the original function, and vice versa.
• Domain: we have for all x that
so
which means the range of
, and hence the domain of
, is
• Range: we get this immediately from the domain restriction we chose earlier,
See the attached plot -
is shown as a dashed orange curve;
with the restricted domain is shown in blue; the inverse
corresponding to the restricted
is shown in green; the other inverse is shown with a dashed red curve. You can see a geometrical properties of inverses: if you mirror the blue curve along the dotted line (y = x), you would get the green curve.