#1**0 **

Well, let's say y is v(t). It'll make it simpler, or at least... it will for me. :)

Then the equation is now y=100(1-t/40)^2

To find the inverse you need to switch the y and x, or in this case, y and t.

That would make **t=100(1-y/40)^2**

Solve for y

First divide both sides by 100 to make **t/100=(1-y/40)^2**

Then square root both sides to get √(t/100)=1-y/40, which simplifies to **√(t)/10=1-y/40.**

Then subtract 1 to both sides to get **-1+√(t)/10=-y/40.**

Next, multiply -1 to both sides to get **1-√(t)/10=y/40**

Then multiply 40 to both sides to get **40-4√(t)=y**

That is **y=40-4√(t)**

Which is **v(t)=40-4√(t) ** because I had used y instead of v(t).

And that's the answer

**v(t)=40-4√(t)**

Correct me if I'm wrong.

Gh0sty15 Aug 23, 2017

#2**+2 **

Substituting x for t, this function is not one-to-one....thus....it has no inverse unless we restrict its domain

y = 100 ( 1 - t/40 )^2 divide both sides by 100

y / 100 = ( 1 - t/40)^2 take both roots of sides

±√ (y/100) = 1 - t/40 multiply both sides by -1

±√ ( y / 100 ) = t / 40 - 1 add 1 to both sides

1 ± √ ( y / 100 ) = t/40 multiply both sides by 40

40 ± 40 √ ( y / 100 ) = t swap t and y

40 ± 40 √ ( t / 100 ) = y for y, write f^{-1}(t)

40 ± 40 √ ( t / 100 ) = f^{-1}(t)

Here is the graph : https://www.desmos.com/calculator/lf5zoe7lei

If we restrict the original domain to ( -infinity, 40), the inverse is represented by the orange graph

If we restrict the original domain to (40, infinity), the inverse is represented by the red graph

CPhill Aug 23, 2017