Pregunta
The function
is one-to-one.
(a) Find its inverse and check your answer. (b) Find the domain and the range of
and
.
(a)
(Simplify your answer.)
(a) Find its inverse and check your answer. (b) Find the domain and the range of
(a)
Ask by Lee Howell. in the United States
Mar 16,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
(a) The inverse function is
.
(b)
-
Domain and Range of
: - Domain: All real numbers except
. - Range: All real numbers except
.
- Domain: All real numbers except
-
Domain and Range of
: - Domain: All real numbers except
. - Range: All real numbers except
.
- Domain: All real numbers except
Solución
Step 1. Set up the equation
We start with the function
Let
, so
To find the inverse function, we swap
and
:
Step 2. Solve for
Multiply both sides by
to eliminate the denominator:
Expand the left side:
Bring the terms involving
to one side:
Factor
on the left side:
Solve for
by dividing both sides by
(note that
):
Thus, the inverse function is
Step 3. Check the Inverse
To check, we compute
:
Start with
Simplify the numerator:
Simplify the denominator:
Now, the whole expression becomes
Since we obtain
, the inverse function is verified.
Step 4. Domain and Range
-
For
: -
Domain of
: The denominator cannot be zero, soTherefore, the domain ofis -
Range of
: As, the function approaches the horizontal asymptote To check ifis achieved, set: Multiply both sides by: This simplifies to:which is impossible. Thus,is not included in the range. Therefore, the range ofis
-
-
For the inverse
: -
Domain of
: The denominator requires thatThus, the domain ofis Note that this is exactly the range of. -
Range of
: The range ofis the domain of , so:
-
Final Answers:
(a) The inverse function is
(b) The domain and range are:
-
For
: - Domain:
- Range:
- Domain:
-
For
: - Domain:
- Range:
- Domain:
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Beyond the Answer
To find the inverse function of
, we start by letting
:
Next, we swap
and
to solve for
:
Now, we will cross-multiply and rearrange:
Rearranging gives:
Factoring out
,
Thus,
So, the inverse function is
To check the inverse, we can find
and verify if it equals
:
Calculating:
This will simplify to
, confirming it as the inverse.
Next, to find the domain and range of
:
- Domain of
: The function is undefined when the denominator is zero, i.e., leading to . Thus, the domain is . - Range of
: Analyzing the horizontal asymptote will help us find this. As approaches infinity, approaches 5. Hence, the range is all real numbers except 5, so .
Now, for the inverse function:
- Domain of
(which corresponds to the range of ): . - Range of
(which corresponds to the domain of ): .
In conclusion:
(a)
(a)
Domain and range:
- Domain of
: - Range of
: - Domain of
: - Range of
:
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