SOLUTION: 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

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].

We go in steps with our transformations:

Note that [a,b] is short for {x∈R|a ≤ x ≤ b}.

Multiplying x by a constant, adding to or subtracting from x -- affects the
domain only, and the range stays the same.

Multiplying a function by a constant, adding to or subtracting from a
function -- affects the range only, and the domain stays the same.

          function        transformation        domain       range
-------------------------------------------------------------------
1.         f(x)                none             [-4,16]      [-8,12]
2.         f(-x)        reflects in y-axis      [-16,4]      [-8,12]
3.         f(-(x-3))   shifts 3 units right     [-13,7]      [-8,12]
           f(-x+3)   (simplification of above)
4.    (3/4)f(-x+3)     shrinks by 3/4 factor    [-13,7]      [-6,9]
5.    (3/4)f(-x+3)+5   shifts 5 units up        [-13,7]      [-1,14]

The domain of g(x) is {x∈R|−13 ≤ x ≤ 7}
The range of g(x) is {y∈R|−1 ≤ x ≤ 14}

Edwin