From the graphs of the hyperbolic functions, we see that all of them are one-to-one except
and
If we restrict the domains of these two functions to the interval
then all the hyperbolic functions are one-to-one, and we can define the
inverse hyperbolic functions . Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.
Definition
Inverse Hyperbolic Functions
Let’s look at how to derive the first equation. The others follow similarly. Suppose
Then,
and, by the definition of the hyperbolic sine function,
Therefore,
Multiplying this equation by
we obtain
This can be solved like a quadratic equation, with the solution
Since
the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that
The exponential function
is increasing if
and decreasing if
Its domain is
and its range is
The logarithmic function
is the inverse of
Its domain is
and its range is
The natural exponential function is
and the natural logarithmic function is
Given an exponential function or logarithmic function in base
we can make a change of base to convert this function to any base
We typically convert to base
The hyperbolic functions involve combinations of the exponential functions
and
As a result, the inverse hyperbolic functions involve the natural logarithm.
For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.
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Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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