Combining Transformations

Question

(1 point) The domain of f(x) is -5≤ x ≤-1. and its range is -2 ≤ f(x) ≤ 5 Determine the domain and range of the following. Submit your answers in interval notation. 

a) f(2x)+4

Domain: 

Range: 

b) 3f(x+4)

Domain: 

Range: 

c) -f(-x)

Domain: 

Range:

Answer 1

Hi,

a) y = f(2x)+4

Domain: first of all you need to know the 4 doesn't affect on domain for finding domain it is enough you solve inequality such as:

You know----> -5 ≤ x ≤ -1 now replace 2x in this inequality

-5 ≤2x ≤ -1 ------- > both side divided by 2

-2.5 ≤ x ≤ -.5 is domain of y = f(2x) + 4

Range: you know -2 ≤ f(x) ≤5 for finding range of new function it is enough add 4 to the given range

-2 + 4 ≤ f(2x) +4 ≤ 5+ 4 It is very important you know input of new function (2x) has no effect on range

2 ≤ ≤f(2x) +4 ≤ 9 so range of y = f(2x) + 4 is [2,9]

After this explanation for new function such as b) y = 3f(x+4)

Domain: -5 ≤ x + 4 ≤-1 -----. -9 ≤ x ≤ -5

Range: you know -2 ≤ f(x) ≤5 the range of new function 3f( x+ 4) must be [ -6 , 15]

for last question c) y = -f(-x)

Domain: -5 ≤ -x ≤-1 -------> 1 ≤ x ≤ 5

Range: we know -2 ≤ f(x) ≤5 times -1 then we have

-5 ≤ -f(x) ≤ 2

NOTE: when any inequality times or divided by negative number must be change ≤ to ≥ or ≥ to ≤

I hope it is useful,

Minoo