Question

Given the function
`h:x=px-(5)/(2)` and the inverse function
`h^(-1) :x=q+2x`, where p and q are constants, find a) the value of p and q c)
`h^(-1) h(-3)`

Answer 1

`p = (1)/(2)`

`q = 5`

`h^(-1)(h(3)) = 3`

Explanation:

Given

`h(x) = px - (5)/(2)`

`h^(-1)(x) = q + 2x`

Solving for p and q

Replace h(x) with y in
`h(x) = px - (5)/(2)`

`y = px - (5)/(2)`

Swap the position of y and d

`x = py - (5)/(2)`

Make y the subject of formula

`py = x + (5)/(2)`

Divide through by p

`y = (x)/(p) + (5)/(2p)`

Now, we've solved for the inverse of h(x);

Replace y with
`h^(-1)(x)`

`h^(-1)(x) = (x)/(p) + (5)/(2p)`

Compare this with
`h^(-1)(x) = q + 2x`

We have that

`(x)/(p) + (5)/(2p) = q + 2x`

By direct comparison

`(x)/(p) = 2x` --- Equation 1

`(5)/(2p) = q` --- Equation 2

Solving equation 1

`(x)/(p) = 2x`

Divide both sides by x

`(1)/(p) = 2`

Take inverse of both sides

`p = (1)/(2)`

Substitute
`p = (1)/(2)` in equation 2

`(5)/(2 * (1)/(2)) = q`

`(5)/(1) = q`

`5 = q`

`q = 5`

Hence, the values of p and q are:
`p = (1)/(2)`;
`q = 5`

Solving for
`h^(-1)(h(3))`

First, we'll solve for h(3) using
`h(x) = px - (5)/(2)`

Substitute
`p = (1)/(2)`; and
`x = 3`

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

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

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

`h(3) = (-2)/(2)`

`h(3) = -1`

So;
`h^(-1)(h(3))` becomes

`h^(-1)(-1)`

Solving for
`h^(-1)(-1)` using
`h^(-1)(x) = q + 2x`

Substitute
`q = 5` and
`x = -1`

`h^(-1)(x) = q + 2x` becomes

`h^(-1)(-1) = 5 + 2 * -1`

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

`h^(-1)(-1) = 3`

Hence;

`h^(-1)(h(3)) = 3`