Question

Find the inverse of the function.
f(x)=9x², x ≤ 0

Answer 1

The inverse of a function "swaps" the x and y values of the original function. To find the inverse of f(x) = 9x², we need to switch the x and y values and solve for x in terms of y:

y = 9x²

To find the inverse function we must isolate x.

x = √(y/9)

So the inverse of the function f(x) = 9x², x ≤ 0 is f^-1(x) = √(x/9), x ≤ 0.

Answer 2

`f^(-1)(x)=(√(x))/(3), \quad x \leq0`

Explanation:

Given function:

`f(x)=9x^2`

The domain of the given function is restricted: {x : x ≤ 0}

The range of the given function is restricted: {f(x) : f(x) ≤ 0}

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

`\implies x=9y^2`

Rearrange the equation to make y the subject:

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

`\implies x=(3y)^2`

`\implies \pm√(x)=3y`

`\implies y=(\pm√(x))/(3)`

As x ≤ 0 then:

`\implies y=(√(x))/(3)`

Replace y with f⁻¹(x):

`\implies f^(-1)(x)=(√(x))/(3)`

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 : x ≤ 0}.