# the one-to-one functions g and h are defined as follows. g={(-4, -5), (1, 2), (7,1), (8, 5)} find the following: g^-1(1)= ----- h^-1(x)= (h * h^-1) (5)=

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In this question we need to find `g^(-1)` and `h^(-1)` , which are the inverse functions of the functions g and h, respectively.

The function g is given as a set of ordered pairs. To create inverse function `g^(-1)` , simply exchange x and y, or first and second coordinate, for each pair:

`g^(-1) = {(-5, -4), (2, 1), (1, 7), (5, 8)}`

Now we see that the value of `g^(-1)` of 1 is the second coordinate of the pair where the first coordinate is 1:

`g^(-1) (1) = 7`

Function h is given as a formula: h(x) = 2x + 3. To find inverse function, exchange x and y in the equation y = 2x + 3 and solve for y:

x= 2y + 3

2y = x - 3

`y = (x-3)/2`

This is the inverse function of h(x):

`h^(-1)(x) = (x-3)/2`

The last expression is a composition of function h and its inverse:

`(h*h^(-1))(5) = h(h^(-1)(5))`

First, evaluate the innermost parenthesis:

`h^(-1)(5) = (5-3)/2 = 1`

Then, evaluate h of the result: h(1) = 2*1 + 3 = 5.

So `(h*h^(-1))(5) = 5`

This result can also be obtain by using the fact that the composition of the function and its inverse is an identity function. That is, applying the function to a value of x, and then applying the inverse will "undo" the function, resulting in the original value of x: `(h*h^(-1))(x) = x`