Lesson Special advanced problems on finding the domain of functions
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<H2>Special advanced problems on finding the domain of functions</H2> <H3>Problem 1</H3>Suppose the domain of f is (-1,3). Define the function g by g(x) = 5 - f(x) + f(5/x). What is the domain of g? <B>Solution</B> <pre> Let D be the domain of f(x), D = (-1,3). Both x and 5/x should be in the interval (-1,3), and, additionally, x should not be equal to 0 (zero). So, the constraints are -1 < x < 3, (1) -1 < 5/x < 3, (2) x =/= 0. (3) Inequalities (1) and (3) tell us that we should consider two separate intervals (-1,0) and (0,3) for x, and determine other limitations on x from inequality (2) (a) So, let x be in the interval (-1,0). Thus, x is negative now. Then first of the two inequalities (2), -1 < 5/x, is equivalent to -x > 5 (after multiplying both sides by negative value of x and flipping the inequality sign), or, which is the same, x < -5. Thus we determined that if x is in (-1,0), then due to first inequality (2), x must be lesser than -5, which is out of the domain D. So, we may exclude this case "x is in (-1,0)" from our consideration. (b) Now, let x be in the interval (0,3). Thus, x is positive now. Then first of the two inequalities (2), -1 < 5/x, is always valid and does not imply other restrictions on x. The second of the two inequalities (2), 5/x < 3, then implies x > 5/3. Thus, if x is positive, then due to second inequality of (2), x must be greater than 5/3. Combining what we found in (a) and (b), the answer to the problem's question is +-------------------------------------------------+ | The domain of the function g(x) is (5/3,3). | +-------------------------------------------------+ </pre> My other lessons in this site on plotting and analyzing functions are - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-x-intercepts-and-y-intercepts.lesson>Finding x-intercepts and y-intercepts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Compressing-and-stretching-of-graphs.lesson>Compressing and stretching graphs</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-functions.lesson>HOW TO PLOT transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-write-functions-for-transformed-plots.lesson>HOW TO write functions for transformed plots</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/HOW-TO-PLOT-transformed-periodic-trigonometry-functions.lesson>HOW TO PLOT transformed periodic trigonometry functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Analyzing-periodic-trig-functions-for-amplitude-period-vert-and-hor-shifts.lesson>Analyzing periodic trigonometric functions for the amplitude, the period, vertical and horizontal shifts</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Do-not-fall-into-a-TRAP-when-analysing-problems-on-trigonometric-functions.lesson>Do not fall into a TRAP when analyzing problems on trigonometric functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/The-domain-and-the-range-of-transformed-functions.lesson>The domain and the range of transformed functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Write-a-function-that-has-given-transformations-from-the-parent-function.lesson>Write a function which is a result of given transformations of the parent function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Describe-transformations-from-the-given-basic-function-to-final-function.lesson>Describe transformations from the given parent function to final function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Writing-a-function-rule-for-a-function-based-on-its-wording-description.lesson>Writing a function rule for a function based on its wording description</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Constructing-a-function-based-on-its-given-properties.lesson>Constructing a function based on its given properties</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Finding-inverse-functions.lesson>Finding inverse functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Miscellaneous-problems-on-plots-of-functions.lesson>Miscellaneous problems on plots of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Given-a-point-on-a-plot-of-a-function--find-the-corresponding-point-on-the-plot-of-transformed-function.lesson>Given a point on a plot of a function, find the corresponding point on the plot of transformed function</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/Special-advanced-problems-on-finding-the-range-of-functions.lesson>Special advanced problems on finding the range of functions</A> - <A HREF=https://www.algebra.com/algebra/homework/Coordinate-system/OVERVIEW-of-lessons-on-plotting-and-analyzing-functions.lesson>OVERVIEW of lessons on plotting and analyzing functions</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
