Math Question ?

Question

Let y=f(x) be a function with domain D =(-inf,inf) and range R= [-5,inf). Find the domain D and range R for the following functions and enter your answers using interval notation. Keep in mind the order of operations. Be sure your intervals are in the correct order, and enter exact answers only.

a. y=-f(2x)-2
Domain= (-inf,inf)
Range= ?

When solving I got the answer 3 so I put [3,inf) and it was said to be wrong.

b. y=2/3f(x-4)-1
Domain: (-inf,inf)
Range: 2/3(-5)-1=-13/3
I put [-13/3,inf), it was also wrong

c. y=-3f(-1/3x)
Domain: (-inf,inf)
Range: 3(-5)=-15
I put [-15,inf), was also wrong.

What am I doing wrong?

Thanks in advance!

az_lender · Accepted Answer

(a)  The 2 inside the parentheses will have no effect on the range.  The minus sign in front of the f will turn the range upside down, so it becomes (-infinity,5].  The -2 at the end will change that to (-infinity, 3].

(b)  The -4 is just a horizontal shift, no effect on the range.  If the (2/3) is a coefficient in front of the f, that would change the range to [-10/3, infinity).  And then the -1 changes it to [-13/3, infinity).  So I agree with your answer in this case.

(c)  The -3 not only magnifies the range by a factor of 3 but also turns it upside down, so that it's (-infinity, +15].

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