Question

inverse functions linear discrete

Answer 1

`\text{g}^(-1)(3) =\boxed{-3}`

`h^(-1)(x)=\boxed{7x+10}`

`\left(h \circ h^(-1)\right)(-2)=\boxed{-2}`

Explanation:

The inverse of a one-to-one function is obtained by reflecting the original function across the line y = x, which swaps the input and output values of the function. Therefore, (x, y) → (y, x).

Given the one-to-one function g is defined as:

`\text{g}=\left\{(-8,8),(-3,3),(3,0),(5,6)\right\}`

Then, the inverse of g is defined as:

`\text{g}^(-1)=\left\{(8,-8),(3,-3),(0,3),(6,5)\right\}`

Therefore, g⁻¹(3) = -3.

`\hrulefill`

To find the inverse of function h(x), begin by replacing h(x) with y:

`y=(x-10)/(7)`

Swap x and y:

`x=(y-10)/(7)`

Rearrange to isolate y:

`\begin{aligned}x&=(y-10)/(7)\\\\7 \cdot x&=7 \cdot (y-10)/(7)\\\\7x&=y-10\\\\y-10&=7x\\\\y-10+10&=7x+10\\\\y&=7x+10\end{aligned}`

Replace y with h⁻¹(x):

`\boxed{h^(-1)(x)=7x+10}`

`\hrulefill`

As h and h⁻¹ are true inverse functions of each other, the composite function (h o h⁻¹)(x) will always yield x. Therefore, (h o h⁻¹)(-2) = -2.

To prove this algebraically, calculate the inverse function of h at the input value x = -2, and then evaluate the original function h at the result.

`\begin{aligned}\left(h \circ h^(-1)\right)(-2)&=h\left[h^(-1)(-2)\right]\\\\&=h\left[7(-2)+10\right]\\\\&=h[-4]\\\\&=((-4)-10)/(7)\\\\&=(-14)/(7)\\\\&=-2\end{aligned}`

Hence proving that (h o h⁻¹)(-2) = -2.