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Given a function: ๐‘“(๐‘ฅ) = ๐‘ฅ2 + 2.a. Does ๐‘“(๐‘ฅ) has an inverse for all values of ๐‘ฅ

Question

Given a function:

f(x)=x2+2 f(x) = x^2 + 2
Does f(x) f(x) have an inverse for all values of x x ?

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Solution

The function f(x) = x^2 + 2 does not have an inverse for all values of x. This is because the function is not one-to-one, meaning that there are multiple values of x that can produce the same output.

For a function to have an inverse, it must be both one-to-one (each x-value corresponds to exactly one y-value) and onto (each y-value corresponds to at least one x-value). The function f(x) = x^2 + 2 is not one-to-one because, for example, both x = 2 and x = -2 produce the same output (y = 6).

Therefore, the function f(x) = x^2 + 2 does not have an inverse for all values of x.

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