Question

Question #d2a0e

Answer 1

See below

Explanation:

`f(t) = 1/(sqrt(t+1))`
`f(y^2-1) = 1/(sqrt((y^2-1)+1))` (Replaced t by `y^2-1`

Simplifying further

`f(y^2-1) = 1/(pmy)`

`f(w-1)^2 = 1/(sqrt((w-1)^2+1))` (Replaced t by `(w-1)^2`

Simplifying further

`f(w-1)^2 = 1/(sqrt(w(w-2)+2)`

Answer 2

See explanation.

Explanation:

`color(blue)("Preamble")`

`f` is just a name given to a particular expression construction

The part in the brackets is what is used in that construction at designated points.

`f("something") =1/sqrt("something" + 1)`

Suppose we have `s` then

`f(s)=1/sqrt(s + 1)`

Suppose we have `b` then

`f(b)=1/sqrt(b + 1)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

`f(color(green)(y^2-1))=1/sqrt(color(green)(y^2-1) + 1) = 1/sqrt(color(magenta)(y^2))=1/(+-y)`

Note that `color(magenta)(y^2)` will be a positive value but the square root of a positive value can be either positive or negative
.....................................................................................................

`f( color(red)([w-1]^2)) = 1/sqrt(color(red)([w-1]^2)+1) =1/sqrt(w^2-2w+2)`

Note that `(-1)^2=+1`