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r-ruslan [8.4K]

4 years ago

15

Identify the solution set of the inequality using the given replacement set.

1 answer:

LenKa [72]4 years ago

5 0

$-10 < -7 \to true \\
-7.2 < -7 \to true \\
-7 < -7 \to not \ true \\
-6.9 < -7 \to not \ true \\
6.5 < -7 \to not \ true \\
9 < -7 \to not \ true \\ \\
\{-10,-7.2 \} \Rightarrow \hbox{the answer is A}$

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If G^-1(x) is the inverse of G(x), what is the value of G^-1(G(x))?

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$G^{-1}(G(x))=x$

Step-by-step explanation:

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Proof in explanation.

Step-by-step explanation:

I'm going to attempt this by squeeze theorem.

We know that $\cos(\frac{2}{x})$ is a variable number between -1 and 1 (inclusive).

This means that $-1 \le \cos(\frac{2}{x}) \le 1$ .

$x^4 \ge 0$ for all value $x$ . So if we multiply all sides of our inequality by this, it will not effect the direction of the inequalities.

$-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4$

By squeeze theorem, if $-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4$

and $\lim_{x \rightarrow 0}-x^4=\lim_{x \rightarrow 0}x^4=L$ , then we can also conclude that $\im_{x \rightarrow} x^4\cos(\frac{2}{x})=L$ .

So we can actually evaluate the "if" limits pretty easily since both are continuous and exist at $x=0$ .

$\lim_{x \rightarrow 0}x^4=0^4=0$

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Some people call this sandwich theorem.

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