# 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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### 1 Answer

`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)` **