a. Explanation Given information : The functions f ( x ) = 1 − e x 2 1 − e 1 − x 2 Concept Involved: Domain means the values of x that gives a defined output to the function. Whenever we are trying to find the domain of the function, we need to identify the restrictions on the domain. The domain of a rational function includes all real numbers except those that cause the denominator to equal zero . To find the domain of a function with a rational expression, set the denominator of the expression not equal to zero and solve for x using the zero product property which states that if a ⋅ b = 0 , then a = 0 or b = 0 . Calculation: Description Steps To find the restriction on the domain set the denominator not equal to zero and solve for x 1 − e 1 − x 2 ≠ 0 Subtract 1 on both sides of the equation 1 − e 1 − x 2 − 1 ≠ 0 − 1 Dividing by -1 on both sides and simplifying the fraction − e 1 − x 2 ≠ − 1 − e 1 − x 2 − 1 ≠ − 1 − 1 e 1 − x 2 ≠ 1 Taking natural logarithm on both sides ln e 1 − x 2 ≠ ln 1 Description Steps Applying the logarithmic property ln m n = n ln m in left side of the equation. Substituting the value of ln e = 1 & ln 1 = 0 ( 1 − x 2 ) ( ln e ) ≠ ln 1 ( 1 − x 2 ) ( 1 ) ≠ 0 1 − x 2 ≠ 0 Factor the left side of the equation 1 2 − x 2 ≠ 0 ( 1 − x ) ( 1 + x ) & b. To determine To find : the domain of the function

15th Edition

ISBN: 9781133109631

Author: Stewart, JAMES, Day, Troy

Publisher: Cengage Learning,

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Chapter 1.4, Problem 19E

To determine

Tofind : the domain of the function

To determine

To find : the domain of the function

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Chapter 1.4, Problem 18E

Chapter 1.4, Problem 20E

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Question 1: If a barometer were built using oil (p = 0.92 g/cm³) instead of mercury (p = 13.6 g/cm³), would the column of oil be higher than, lower than, or the same as the column of mercury at 1.00 atm? If the level is different, by what factor? Explain. (5 pts) Solution: A barometer works based on the principle that the pressure exerted by the liquid column balances atmospheric pressure. The pressure is given by: P = pgh Since the atmospheric pressure remains constant (P = 1.00 atm), the height of the liquid column is inversely proportional to its density: Step 1: Given Data PHg hol=hgx Poil • Density of mercury: PHg = 13.6 g/cm³ Density of oil: Poil = 0.92 g/cm³ • Standard height of mercury at 1.00 atm: hμg Step 2: Compute Height of Oil = 760 mm = 0.760 m 13.6 hoil = 0.760 x 0.92 hoil = 0.760 × 14.78 hoil = 11.23 m Step 3: Compare Heights Since oil is less dense than mercury, the column of oil must be much taller than that of mercury. The factor by which it is taller is: Final…

Chapter 1 Solutions

Biocalculus

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Tofind : the domain of the function

Solution Summary: The author explains that the domain of a rational function includes all real numbers except those that cause the denominator to equal zero.

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Chapter 1 Solutions