Question 153021
# 1)


{{{y=3+x^3}}} Start with the given equation.



{{{x=3+y^3}}} Switch x and y



{{{x-3=y^3}}} Subtract 3 from both sides.



{{{root(3,x-3)=y}}} Subtract 3 from both sides.



{{{y=root(3,x-3)}}}



So the inverse function is *[Tex \LARGE f^{-1}\left(x\right)=\sqrt[3]{x-3}]



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# 2)


<h4>Positive Zeros:</h4>


First count the sign changes of {{{f(x)=2x^4+3x^3-2x^2+x-2}}}


From {{{2x^4}}} to {{{3x^3}}}, there is no change in sign


From {{{3x^3}}} to {{{-2x^2}}}, there is a sign change from positive to negative 


From {{{-2x^2}}} to {{{x}}}, there is a sign change from negative to positive 


From {{{x}}} to {{{-2}}}, there is a sign change from positive to negative 


So there are 3 sign changes for the expression {{{f(x)=2x^4+3x^3-2x^2+x-2}}}. 


So there are 3 or 1 positive zeros




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<h4>Negative Zeros:</h4>


{{{f(-x)=2(-x)^4+3(-x)^3-2(-x)^2+(-x)-2}}} Now let's replace each {{{x}}} with {{{-x}}}



{{{f(-x)=2x^4-3x^3-2x^2-x-2}}} Simplify



Now let's count the sign changes of {{{f(-x)=2x^4-3x^3-2x^2-x-2}}}


From {{{2x^4}}} to {{{-3x^3}}}, there is a sign change from positive to negative 


From {{{-3x^3}}} to {{{-2x^2}}}, there is no change in sign


From {{{-2x^2}}} to {{{-x}}}, there is no change in sign


From {{{-x}}} to {{{-2}}}, there is no change in sign


So there is 1 sign change for the expression {{{f(-x)=2x^4-3x^3-2x^2-x-2}}}. 


So there is 1 negative zero