Question

Restrict the domain of the quadratic function and find its inverse. Confirm the inverse relationship using composition. f(x) = 0.2x2 The domain is (x| x ). Since f−1f(x) = x for x 0, it has been confirmed that f−1(x) = for x ≥ 0 is the inverse function of f(x) = 0.2x2 for x 0. Chose the graph of the function and its inverse.

Answer 1

Domain = x≥0

Inverse = f⁻¹(x) =
`\sqrt 5y`.

Explanation:

The given function is
`f(x)=0.2x^(2)`

We know that the domain of all functions is the whole real line.

But as
`x^(2)\geq 0`.

So, the domain of f(x) is the positive real line.

Thus, the restriction to the domain of f(x) is x≥0 .

Now, we will find the inverse of f(x),

`y=0.2x^(2)`

i.e.
`x^(2)=(y)/(0.2)`

i.e.
`x^(2)=5y`

i.e.
`x=\sqrt 5y`

Hence, the inverse of f(x) is f⁻¹(x) =
`\sqrt 5y`.

Further, we will check the inverse using composition rule.

i.e. fοf⁻¹(x) = f⁻¹οf(x)

i.e. f(f⁻¹(x)) = f⁻¹(f(x))

i.e. f(
`\sqrt 5y`) = f⁻¹(
`0.2x^(2)`)

i.e.
`0.2(√(5x))^(2)` =
`\sqrt{5* 0.2x^(2)}`

i.e. 0.2 × 5x =
`\sqrt{x^(2)}`

i.e. x = x

Hence, we get that the function
`f(x)=0.2x^(2)` has inverse f⁻¹(x) =
`\sqrt 5y`.

The graph of the function and its inverse can be seen below.

Answer 2

The given function is

f(x)= 0.2 x²

Since f(x) will be defined for all real values of x.

So, Domain of f(x) will be ( x| x is a real number.)→This is set builder notation.

Finding the inverse of f(x):

y = 0.2 x²

→ x²= 5 y

→x =
`\pm√(5 y)`→ → Inverse of f(x)

Replacing x by y and y by x,we get inverse of the given function

y =
`\pm√(5 x)`→ →Domain x ≥ 0, x∈[0,∞]

Graph of function and its inverse are shown below.