For the function f defined by `f(x)=(4x)/(x^2+6)` , solve f(-4) , f(-x) , -f(x), and f(x+h) .  

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lemjay | High School Teacher | (Level 3) Senior Educator

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`f(x)=(4x)/(x^2+6)`

> To determine f(-4), replace x with -4.

`f(-4) = (4*(-4))/((-4)^2+6) = (-16)/(16+6)= -16/22`

The, reduce the fraction to its lowest term.

`f(-4)=-8/11`

> To solve for f(-x), multiply the x variable by -1. Or we may simply replace x with -x. Since `-1*x=-x` .

`f(-x)=(4*(-x))/((-x)^2+6)= (-4x)/(x^2+6)`

`f(-x)=-(4x)/(x^2+6)`

> To determine -f(x), multiply the both sides of the given function by -1. Note that -f(x) is the same as `-1*f(x)` .

`-1*f(x)= (4x)/(x^2+6)*(-1)`

`-f(x)= -(4x)/(x^2+6)`

> For f(x+h), replace x with x+h.

`f(x+h) = (4(x+h))/((x+h)^2+6) =(4x+4h)/(x^2+xh+xh+h^2+6)`

`f(x+h) = (4x+4h)/(x^2+2xh+h^2+6)`

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Hence, 

     ` f(-4) = -8/11`  ,                            `f(-x) = -(4x)/(x^2+6)`

     ` -f(x)=-(4x)/(x^2+6)`         and        `f(x+h)= (4x+4h)/(x^2+2xh+h^2+6)`        

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