Functions f and g are defined by f(x)=3/x and g(x)=2x+1.Express x=(x-1)/2 in terms of f or/and g

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to express the equation `x=(x-1)/2`  in terms of `f(x) = 3/x`  and `g(x) = 2x + 1`  such that:

`x=(x-1)/2`

You need to multiply by 2 both sides such that:

`2x = (2(x-1))/2 `

Reducing duplicate factors yields:

`2x = x - 1 `

You need to add 1 both sides, such that:

`2x + 1 = x`

Notice that the equation `2x + 1`  is the equation of the function `g(x) = 2x + 1` , hence, you may substitute `g(x)`  for `2x + 1`  such that:

`g(x) = x`

You need to divide by 3 both sides, such that:

`g(x)/3 = x/3`

Notice that the function `f(x) = 3/x` , hence `x/3` `= 1/(f(x)), ` thus, you may substitute `1/(f(x))`  for `x/3`  such that:

`g(x)/3 = 1/(f(x))`

You need to perform cross mutliplication such that:

`3*1 = f(x)g(x)`

Hence, using the equation of the functions `f(x)`  and `g(x)`  yields that you may write the expression `x=(x-1)/2`  in terms of `f(x)`  and `g(x)`  such that  `f(x)g(x) = 3` .

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vaaruni | High School Teacher | (Level 1) Salutatorian

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Given f(x)=3/x   and    g(x)= 2x + 1

 he given equation  =>   x = (x-1)/2                

                            Or,   2x = x -1                                     

                            Or,   2x + 1 = x                 

                            Or,  g(x) = 3/(fx)   [ since, g(x)=2x+1,  x= 3/)f(x) ]

                            Or,   g(x)*f(x)=3

                        Hence,   g(x)*f(x) = 3    <--- Answer   

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vaaruni | High School Teacher | (Level 1) Salutatorian

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Given : f(x) = 3/x     and  g(x) = 2x + 1   

 f(x) = 3/x   

 Or,    x = 3/f(x)      -------- (1)  And   g(x)= 2x + 1  -----(2)

Given equation : 

       x = (x - 1)/2

Or,   2x = x - 1

Or,   2x + 1 = x

Or,   g(x) = 3/f(x)        [ substituting value of 2x+1 from eq.(1)                                     and value  of x  from eq.(2) ]        

 Or,   g(x).f(x) = 3   

 Or,    f(x).g(x) = 3

Hence x = (x - 1)/2   is equivalent to  :

 f(x).g(x)= 3  Answer

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