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# pls help!!!!! due soon

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Let f(x)= |x| and g(x)=x^2

Find all values of x for which f(x)>g(x)

Mar 30, 2024

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To find all values of $$x$$ for which $$f(x) > g(x)$$, where $$f(x) = |x|$$ and $$g(x) = x^2$$, we need to compare their values for different ranges of $$x$$.

1. For $$x \geq 0$$:

$f(x) = |x| = x$

$g(x) = x^2$

So, $$f(x) > g(x)$$ for $$x \geq 0$$ when $$x^2 > x$$, which is true when $$x > 1$$ or $$x < 0$$.

2. For $$x < 0$$:

$f(x) = |x| = -x$

$g(x) = x^2$

So, $$f(x) > g(x)$$ for $$x < 0$$ when $$-x > x^2$$, which is true when $$-x > x^2$$ for $$x < 0$$.

Let's solve $$-x > x^2$$ for $$x < 0$$:

$-x > x^2$

$x^2 + x < 0$

$x(x + 1) < 0$

This inequality holds true when either $$x < 0$$ and $$x + 1 > 0$$, or $$x > 0$$ and $$x + 1 < 0$$.

1. For $$x < 0$$, $$x(x + 1) < 0$$ when $$x < 0$$ and $$x + 1 > 0$$ (the product of two negative numbers is positive):

$x < 0$

$x + 1 > 0$

$x > -1$

So, for $$x < 0$$, $$f(x) > g(x)$$ when $$-1 < x < 0$$.

Combining both ranges of $$x$$, we have $$f(x) > g(x)$$ for $$x > 1$$ or $$-1 < x < 0$$.

Therefore, all values of $$x$$ for which $$f(x) > g(x)$$ are $$x \in (-1, 0) \cup (1, \infty)$$.

Mar 30, 2024