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# Functions

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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
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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