Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He als
o asked them to rate their mood on a scale from 000 to 101010, with 101010 being the happiest. A line was fit to the data to model the relationship.
2 answers:
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The answer is:
The inverse of a function is another function,
, with the following property:
In other words, the inverse of a function does exactly "the opposite" of what the original function does, and so if you compute them both in sequence you return to the starting point.
Think for example to a function that doubles the input, , and one that halves it:
. Their composition is clearly the identity function
, since you consider "twice the half of something", or "half the double of something".
In general, to invert a function , you have to solve the expression for
, writing an expression like
. If you manage to do so, then
is the inverse of
.
In your case, you have
Multiply both sides by to get
Square both sides to get
Finally, subtract 3 from both sides to get
Since the name of the variables doesn't really have a meaning, you can say that the inverse function is
As for the domain of the inverse function, remember what we said ad the beginning: if the original function goes from set A (domain) to set B (codomain), then the inverse function goes from set B (domain) to set A (codomain). This means that the inverse function is defined on an element in B if and only if that element belongs to the range of the original function, i.e. the set of the elements of the codomain such that there exists
. So, we need the range of
.
We know that the range of is
. When you transform it to
you simply translate the graph horizontally, so the range doesn't change. But when you multiply the function times
you affect both extrema of the range, turning it into
, which you can simply write as