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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

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