Question

Inverse functions linear discrete

Answer 1

`\text{g}^(-1)(x)=\boxed{(x-13)/(2)}`

`\left(\text{g}^(-1) \circ \text{g}\right)(-4)=\boxed{-4}`

`h^(-1)(9)=\boxed{-3}`

Explanation:

To find the inverse of function g(x) = 2x + 13, begin by replacing g(x) with y:

`y=2x+13`

Swap x and y:

`x=2y+13`

Rearrange to isolate y:

`\begin{aligned}x&=2y+13\\\\x-13&=2y+13-13\\\\x-13&=2y\\\\2y&=x-13\\\\(2y)/(2)&=(x-13)/(2)\\\\y&=(x-13)/(2)\end{aligned}`

Replace y with g⁻¹(x):

`\boxed{\text{g}^(-1)(x)=(x-13)/(2)}`

`\hrulefill`

As g and g⁻¹ are true inverse functions of each other, the composite function (g⁻¹ o g)(x) will always yield x. Therefore, (g⁻¹ o g)(-4) = -4.

To prove this algebraically, calculate the original function g at the input value x = -4, and then evaluate the inverse function of g at the result.

`\begin{aligned}\left(\text{g}^(-1) \circ \text{g}\right)(-4)&=\text{g}^(-1)\left[\text{g}(-4)\right]\\\\&=\text{g}^(-1)\left[2(-4)+13\right]\\\\&=\text{g}^(-1)\left[5\right]\\\\&=((5)-13)/(2)\\\\&=(-8)/(2)\\\\&=-4\end{aligned}`

Hence proving that (g⁻¹ o g)(-4) = -4.

`\hrulefill`

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 h is defined as:

`h=\left\{(-3,9), (1,0), (3,-7), (5,2), (9,6)\right\}`

Then, the inverse of h is defined as:

`h^(-1)=\left\{(9,-3),(0,1),(-7,3),(2,5),(6,9)\right\}`

Therefore, h⁻¹(9) = -3.