Question

????The domain of a function ƒ(x) is {xϵℝ/-1<x<5} a)Determine the domain of ƒ(x+5) b)Determine the domain of ƒ(-2x+5)

Answer 1

`a)` The domain of `f(x+5)` is `{x in RR | –6 < x < 0}.`
`b)` The domain of `f(–2x+5)` is `{x in RR | 0 < x < 3}.`

Explanation:

The domain of a function `f` is all the allowable input values. In other words, it's the set of inputs for which `f` knows how to give an output.

If `f(x)` has the domain of `{x in RR | –1 < x < 5}`, that means for any value strictly between –1 and 5, `f` can take that value, "do its magic", and give us a corresponding output. For every other input value, `f` has no idea what to do—the function is undefined outside of its domain.

So, if our function `f` needs its inputs to be strictly between –1 and 5, and we want to give it an input of `x+5`, what are the restrictions on that input expression? We need `x+5` to be strictly between –1 and 5, which we can write as

`–1" "<" "x+5" "<" "5`

This is an inequality which can be simplified (so that `x` is by itself in the middle). Subtracting 5 from all 3 "sides" of the inequality, we get

`–6" "<" "x" "<" "0`

This tells us the domain of `f(x+5)` is `{x in RR | –6 < x < 0}.`

Basically, you just need to just replace the `x` in the domain interval with the new input (argument). Let's illustrate with part b):

`"D"[f(x)]={x in RR | –1 < x < 5}`

means

`"D"[f(color(red)(–2x+5))] = {x in RR | –1 < color(red)(–2x + 5) < 5}`

which is simplified to

`color(white)("D"[f(–2x+5)]) = {x in RR | –6 < –2x < 0}`

`color(white)("D"[f(–2x+5)]) = {x in RR | 3 > x > 0}`

Don't forget to flip the inequality symbols when dividing through by negatives!

So:

`"D"[f(–2x+5)] = {x in RR | 0 < x < 3}`