Suppose f is a function with domain (-10,6] and range (6,8]. Find the domain and range of the following functions.

Question

Hello everyone, I have this question about finding domain and range

"Suppose f is a function with domain (-10,6] and range (6,8]. Find the domain and range of the following functions

f(4 x+2) "

Can anyone explain to me how can I solve this question?

Answer 1

We can think about this in terms of transformations -- shifts, stretches, shrinks, etc.

In this case, f(4x+2) represents 2 horizontal transforms applied to the graph, meaning the range stays as it is, f(x) ε (6 , 8].

We should rewrite the function with the 4 factored out to put it in standard form: f(4(x+1/2))

This expression indicates the graph will first be shrunk by a factor of 4, ie the points will move 4 times closer to the y-axis than they were, or the x-coordinates will be divided by 4. Then, the points are shifted left by 1/2, meaning we subtract 1/2 from the x-coordinates.

Applying these two transforms gives x ε (- 3 , 1] , which is the domain of the new function.

You can verify that applying 4x + 2 to these new endpoints gives the old endpoints of the domain interval.