Lesson The domain and the range of transformed functions

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The domain and the range of transformed functions

Problem 1

The domain of a function f(x) is {x∈R|−4 ≤ x ≤ 16}. The range of f(x) is {y∈R| −8 ≤ y ≤ 12}.
Given g(x) = 3/4f(−x+3) + 5, what is the domain and range of g(x) ?

Solution

As the function g(x) defined via function f(x),


(a)  to find the domain of g(x), we should start from the domain [-4,16]  of f(x), and then first

     make a mirror reflection relative to y-axis, getting the segment [-16,4],

     and then translate this last segment 3 units to the right,

     so the final answer regarding the domain of g(x) is  [-13,7].



(b)  to find the range of g(x),  compress the range of  f(x)  with the compression coefficient  3/4,

     and then shift it 5 units in positive direction.

     By doing this way, you will get the segment [-6,9] after compression and the segment [-1,14] 
     after shifting 5 units in positive direction.



ANSWER.  The domain of g(x) is the segment [-13,7].

         The range of g(x) is the segment [-1.14].

My other lessons in this site on plotting and analyzing functions are
    - Finding x-intercepts and y-intercepts
    - Compressing and stretching graphs
    - HOW TO PLOT transformed functions
    - HOW TO write functions for transformed plots
    - HOW TO PLOT transformed periodic trigonometry functions
    - Analyzing periodic trigonometric functions for the amplitude, the period, vertical and horizontal shifts
    - Do not fall into a TRAP when analyzing problems on trigonometric functions
    - Write a function which is a result of given transformations of the parent function
    - Describe transformations from the given parent function to final function
    - Writing a function rule for a function based on its wording description
    - Constructing a function based on its given properties
    - Finding inverse functions
    - Miscellaneous problems on plots of functions
    - Given a point on a plot of a function, find the corresponding point on the plot of transformed function
    - Special advanced problems on finding the domain of functions
    - Special advanced problems on finding the range of functions
    - OVERVIEW of lessons on plotting and analyzing functions

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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