Question

Let y = f(x) be a function with domain D = [−6, 10] and range R = [−8, 12]. Find the domain D and range R for each function. Assume f(6) = 12 and f(10) = −8. (Enter your answers using interval notation.)
a) y = −2f(x)
b) y = |f(x)|
c) y = f(x + 2) − 3

Answer 1

A) The domain remains the same.
Since, in y = - 2 f(x) , the y is whats being multiplied and therefore changed, not the x  or domain.
Therefore D[-6,10] 
The range is what changes by a factor of -2.
If the original range was [-8,12], then the new one is 
[(-8 x -2),(12 x -2)] = [16 , -24].

However, you must flip the numbers to make the interval true, giving you R [-24,16]

Answer 2

Answer:

a.Domain ,D=[-6,10]

Range ,R=[-24,16]

b.Domain,D=[-6,10]

Range=[8,12]

c.Domain=[-4,12]

Range=[-11,9]

Step-by-step explanation:

We are given that $y=f(x)$  be  a function with domain D=[-6,10] and range R=[-8,12].

Assume f(-6)=12, f(10)=-8

a.$y=-2f(x)$

We have to find the domain D and range R of the function.

Substitute f(-6)=12 then we get

$y=-2(120=-24$

Substitute f(10)=-8[/tex]

$y=-2(-8)=16$

Domain remain same like domain  of given function.

Domain ,D=[-6,10]

Range ,R=[-24,16]

b.$y=\mid f(x)\mid$

We have to find the value of domain and range.

$y=\mid f(-6)\mid=\mid 12\mid=12$

$y=\mid f(10)\mid=\mid -8\mid=8$

Domain,D=[-6,10]

Range=[8,12]

c.$y=f(x+2)-3$

Substitute x=-6 then

$y=f(-6+2)-3=f(-4)-3$

Substitute x=10

Then, we get $y=f(10+2)-3=f(12)-3$

Domain=[-4,12]

f(-6)=12 then , we get

$y=12-3=9$

Substitute x=10

then, we get$y=-8-3=-11$

Range of f(x)=Range f(x+2)

Range=[-11,9]