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VladimirAG [237]
3 years ago
9

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.

Mathematics
2 answers:
ANTONII [103]3 years ago
8 0

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\sqrt{5 y}→ → Inverse of f(x)

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

y = \pm\sqrt{5 x}→ →Domain x ≥ 0, x∈[0,∞]

Graph of function and its inverse are shown below.

Alla [95]3 years ago
8 0

Answer:

Domain = {x | x≥0 }

Inverse = f⁻¹(x) = \sqrt 5y.

Step-by-step 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 | x≥0 }.

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

y=0.2x^{2}

i.e. x^{2}=\frac{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(\sqrt{5x})^{2} = \sqrt{5\times 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.

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