Find the Domain and Range f(x)=((3e^x)/(e^x-19))

Problem

ƒ(x)=(3*ex)/(ex−19)

Solution

1. Identify the domain constraint by setting the denominator not equal to zero, as division by zero is undefined.

ex−19≠0

2. Solve for x by isolating the exponential term and taking the natural logarithm of both sides.

ex≠19

x≠ln(19)

3. State the domain in interval notation based on the restriction found.

Domain: *(−∞,ln(19))∪(ln(19),∞)

4. Find the range by solving the equation y=(3*ex)/(ex−19) for x to determine which values of y are possible.

y*(ex−19)=3*ex

y*ex−19*y=3*ex

5. Isolate the exponential term to solve for x in terms of y

y*ex−3*ex=19*y

ex*(y−3)=19*y

ex=(19*y)/(y−3)

6. Apply logarithmic constraints noting that ex must be strictly positive, so (19*y)/(y−3)>0 and the denominator cannot be zero, so y≠3

(19*y)/(y−3)>0

7. Determine the sign intervals for the inequality. The expression is positive when y<0 or y>3

Range: *(−∞,0)∪(3,∞)

Final Answer

Domain: *(−∞,ln(19))∪(ln(19),∞), Range: *(−∞,0)∪(3,∞)

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