The function f: R to R defined by f(x) = x|x| | vedclass

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(B) Given $f(x) = x|x| + x^3|x|$.We can write this as $f(x) = |x|(x + x^3)$.Case $1$: If $x \geq 0$,then $|x| = x$. So,$f(x) = x(x + x^3) = x^2 + x^4$

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one-one onto

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(B) Given $f(x) = x|x| + x^3|x|$.We can write this as $f(x) = |x|(x + x^3)$.Case $1$: If $x \geq 0$,then $|x| = x$. So,$f(x) = x(x + x^3) = x^2 + x^4$.Case $2$: If $x Thus,$f(x) = \begin{cases} x^4 + x^2, & x \geq 0 \ -(x^4 + x^2), & x For $x \geq 0$,$f'(x) = 4x^3 + 2x > 0$ for $x > 0$. Since $f(0) = 0$,$f(x)$ is strictly increasing for $x \geq 0$.For $x  0$ for $x Since the function is strictly increasing on its entire domain $R$,it is a one-one function.As $x 	o \infty, f(x) 	o \infty$ and as $x 	o -\infty, f(x) 	o -\infty$. Since $f(x)$ is continuous,the range is $(-\infty, \infty) = R$.Since the range equals the codomain,the function is onto.Therefore,the function is one-one onto.

The function $f: R \to R$ defined by $f(x) = x|x| + x^3|x|$ is

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