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 + 4x²

Next replace x with y and y with x.

x = 64 - 32y + 4y²

x = 4(16 - 8y + y²)

x/4 = 16 - 8y + y²

Solving for y, we get,

x/4 = y² - 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.

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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.