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Consider 2 functions f(x), g(x) that is defined on the interval [0, \infty) where
f(x)=x^3+x^2+x+2
g(x)=2f^{-1}(x)-1

Find the value of \(g(0) + g(3)\)

 Apr 15, 2022
 #1
avatar+9459 
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g(0) is not well-defined since f^{-1}(0) does not exist in [0, \infty), i.e., there is no x \in [0, \infty) such that f(x) = 0.

 

Proof:

Since \(x\in [0, \infty)\)\(x^3, x^2, x\) are all nonnegative.

Then \(f(x) = x^3 + x^2 + x + 2 \geq 0 + 0 + 0 + 2 = 2 > 0\) for any \(x\in [0, \infty)\).

Then we have \(f(x) > 0\) for any \(x\in [0, \infty)\).

This implies there are no \(x\in [0, \infty)\) such that \(f(x) = 0\).

 

Therefore, we cannot find the value of \(g(0) + g(3)\).

 Apr 16, 2022

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