Question

I'm doing a practice, and I'm really confused by this question. Help would really be appreciated!!! **100 PTS!**

Answer 1

Or 50

Explanation:

Answer 2

Great Question!

The problem we have at hand is known to be a function, which maps elements from one set of objects, ( the domain ) onto another, the range. If we were to consider an ordered pair, say ( x, y ), then the function would map x onto y. The inverse function is simply the reverse. Take the ordered pair (-4,0). Function g would map - 4 onto 0, such that
`g( - 4 ) = 0`. Therefore, the inverse function would map 0 onto - 4, resulting in
`g^(-1)( 0 ) = - 4`. And there you have it! Our first part is answered!

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This second bit here is interesting. Let
`y = h( x )` -

`y = 4x + 3` - Switch x and y,

`x = 4y + 3` - And now solve this equation for y,

`x - 4y = 3,\\- 4y = - x + 3,\\y = 1 / 4x - 3 / 4`

As you can see, we have taken the inverse of h( x ). As y = h( x ), we can thus conclude the following -

`h^(-1)(x) = 1 / 4x - 3 / 4`

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The composition (h^-1 o h)(-5) is, in other words, h^-1(h(-5)). We can therefore calculate h(-5) and then take it's inverse -

`h(-5) = 4(-5) + 3,\\h(-5) = - 20 + 3,\\h(-5) = - 17`

Now we can take it's inverse -

`h^(-1)(-17) = 1 / 4( - 17 ) - 3 / 4,\\- 17 / 4 - 3 / 4,\\= - 5!`

Our solution for this last bit is - 5. And, if you don't feel like reading through this entire explanation just take a look at the " summed up " answer below,

`g^(-1)( 0 ) = - 4,\\\\h^(-1)( x ) = 1 / 4x - 3 / 4,\\\\( h * h^(-1) )( - 5 ) = - 5` I do hope that helps you!