Pregunta
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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.)

Ask by Lee Howell. in the United States
Mar 16,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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(a) The inverse function is .
(b)
  • Domain and Range of :
    • Domain: All real numbers except .
    • Range: All real numbers except .
  • Domain and Range of :
    • Domain: All real numbers except .
    • Range: 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, so
      Therefore, the domain of is
    • Range of :
      As , the function approaches the horizontal asymptote
      To check if is achieved, set:
      Multiply both sides by :
      This simplifies to:
      which is impossible. Thus, is not included in the range.
      Therefore, the range of is
  • For the inverse :
    • Domain of :
      The denominator requires that
      Thus, the domain of is
      Note that this is exactly the range of .
    • Range of :
      The range of is the domain of , so:

Final Answers:
(a) The inverse function is
(b) The domain and range are:
  • For :
    • Domain:
    • Range:
  • For :
    • Domain:
    • Range:

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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 :
  1. Domain of : The function is undefined when the denominator is zero, i.e., leading to . Thus, the domain is .
  2. 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:
  1. Domain of (which corresponds to the range of ): .
  2. Range of (which corresponds to the domain of ): .
In conclusion:
(a)
Domain and range:
  • Domain of :
  • Range of :
  • Domain of :
  • Range of :

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