Question

5x-3/x-1 what is the inverse of this

Answer 1

`f^(-1)(x)=(3-x)/(5-x), \quad \x\\eq 5\`

Explanation:

Given function:

`f(x)=(5x-3)/(x-1)`

As the given function is a rational function, the domain and range are restricted.

• Domain: (-∞, 1) ∪ (1, ∞).

• Range: (-∞, 5) ∪ (5, ∞).

f⁻¹(x) is the notation for the inverse of the function.

The inverse of a function is a reflection in the line y = x.

To find the inverse of the given function, swap f(x) for y:

`\implies y=(5x-3)/(x-1)`

Rearrange the equation to isolate x:

`\implies y(x-1)=5x-3`

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

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

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

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

Swap the x for f⁻¹(x) and the y for x:

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

The range of the function is the domain of the inverse function.

Therefore, the domain of the inverse function is restricted:

• Domain of f⁻¹(x): (-∞, 5) ∪ (5, ∞).