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# For the function f(x) = (8 - 2x)^{2} ,find f^{-1} . Determine whether f-1 is a function.

**Solution:**

An inverse function reverses the operation done by a particular function. In other words, the inverse function undoes the action of the other function.

Given, f(x) = (8 - 2x)^{2}

First replace f(x) with y.

y = (8 - 2x)^{2}

Using the multiplicative distributive property

y = 64 - 32x + 4x^{2}

Next replace x with y and y with x.

x = 64 - 32y + 4y^{2}

x = 4(16 - 8y + y^{2})

x/4 = 16 - 8y + y^{2}

Solving for y, we get,

x/4 = y^{2} - 4y - 4y + 16

x/4 = y(y - 4) - 4(y - 4)

x/4 = (y - 4) (y - 4)

x/4 = (y - 4)^{2}

x = 4(y - 4)^{2}

Taking square root,

√x = 2(y - 4)

√x = 2y - 8

2y = √x + 8

y = (1/2)√x + 4

Finally replace y with f^{-1}(x).

f^{-1} (x) = (1/2)√x + 4

**Verification:**

(f ∘ f^{-1})(x)= x

(f ∘ f^{-1}) (x)= f [f^{-1}(x)]

= f [(1/2)√x + 4]

= (8 - 2[(1/2)√x + 4])^{2}

= (8 - √x + 8)^{2}

= (-√x)^{2}

= x

Therefore, f^{-1}(x) is a function.

## For the function f(x) = (8 - 2x)^{2} ,find f^{-1}. Determine whether f^{-1} is a function.

**Summary:**

For the function f(x) = (8 - 2x)^{2} , f^{-1}(x) = (1/2)√x + 4.

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