given f(x)=2x-5, construct a function g such that g(f(x))=x

Question

 another question: Given f(x)=2x-5, construct a function g such that g(f(g(5)))=3

Answer 1

Hi Xin;

given f(x)=2x-5, construct a function g such that g(f(x))=x

 

g(x)=function such that when we replace x with 2x-5, the result is x.

We would begin with...

g(x)=(1/2)(.......)

(1/2)(2)=1, the coefficient of x.

We now have...

g(x)=(1/2)(x)

g(f(x))=(1/2)(2x-5)=x-2.5

Let's finish with...

g(x)=(1/2)x+2.5

g(f(x))=[(1/2)(2x-5)]+2.5

g(f(x))=x-2.5+2.5

g(f(x))=x

g(x)=(1/2)x+2.5

 

another question: Given f(x)=2x-5, construct a function g such that g(f(g(5)))=3

I think what you mean is f(g(5))=3....Please correct me if I am wrong.

We want...

2x-5=3

[(2)(5)]-5=3

10-5=3

5=3

 

We need 2x reduced by a value of 2.

We need x reduced by a value of 1.

[2(x-1)]-5=3

[2(5-1)]-5=3

[(2)(4)]-5=3

8-5=3

3=3

g(x)=x-1

f(x)=2x-5

f(g(x))=[(2)(x-1)]-5

f(g(5))=[(2)(5-1)]-5=[(2)(4)]-5=8-5=3

 

Unless you mean...

g(f(5))=3

g(x)=x=3

g(f(5))=2x-5=[(2)(5)]-5=10-5=5

We want the result of 3.

g(x)=x-2

g(f(x))=(2x-5)-2=2x-7

g(f(5))=[(2)(5)]-7=10-7=3

Answer 2

f(x)=2x-5

f(x)+5=2x

(f(x)+5)/2=x

Let g(x)=(x+5)/2 so that g(f(x))=(f(x)+5)/2=x

 

f(x)=2x-5

Use guess for g(x)=(x+b)/2

f(g(x))=2g(x)-5=x+b-5

g(f(g(x)))=g(x+b-5)=(x+2b-5)/2

g(f(g(5)))=b=3

g(x)=(x+3)/2

 

Answer 3

f( x ) = 2X -5 

 

  g ( f(x) ) = X

 

  g ( 2X - 5 ) = x 

 

  g(( 2x - 5 + 5) /2 ) = X

 

   g (( f( X ) + 5 ) /2 ) =X