Question

NO LINKS!!!

12. Find the inverse of the following functions using what you know about inverse operations. Some of these functions do not have a functional inverse. If it has no functional inverse, write "No inverse function"

Answer 1

a) No inverse function

`\textsf{b)} \quad c^(-1)(x)=5+√(x-2),\;\; c\geq2`

`\textsf{c)} \quad f^(-1)(x)=-(2x)/(3x-5)`

Explanation:

Part a

Given function:

`a(x)=-4x^2+3`

To find the inverse of the function, swap x and y:

`\implies x=-4y^2+3`

Rearrange the equation to isolate y:

`\implies x-3=-4y^2`

`\implies y^2=-(x-3)/(4)`

`\implies y^2=(3-x)/(4)`

As the domain and range of function a(x) are unrestricted, the inverse relation is a “sideways” parabola and therefore it is not a function, since it fails the vertical line test.

Part b

Given function:

`c(x)=(x-5)^2+2,\;\;x\geq5`

As the domain of function c(x) is restricted to x ≥ 5, the range is also restricted.

• Range: c(x) ≥ 2

To find the inverse of the function, swap x and y:

`\implies x=(y-5)^2+2`

Rearrange the equation to isolate y:

`\implies x-2=(y-5)^2`

`\implies \pm√(x-2)=y-5`

`\implies y=5\pm√(x-2)`

As the domain of function c(x) is restricted to x ≥ 5 then:

`\implies y=5+√(x-2)`

Replace y with c⁻¹(x):

`\implies c^(-1)(x)=5+√(x-2)`

The domain of the inverse of a function is the same as the range of the original function. Therefore, the domain of the inverse function is restricted to x ≥ 2.

Part c

Given function:

`f(x)=(5x)/(3x+2)`

To find the inverse of the function, swap x and y:

`x=(5y)/(3y+2)`

Rearrange the equation to isolate y:

`\implies x(3y+2)=5y`

`\implies 3xy+2x=5y`

`\implies 3xy-5y=-2x`

`\implies y(3x-5)=-2x`

`\implies y=-(2x)/(3x-5)`

Replace y with f⁻¹(x):

`\implies f^(-1)(x)=-(2x)/(3x-5)`