Given the functions f(x)=x+2 and g(x)=x^2, determine all values of x for which f(g(x))=g(f(x))

Question

Given the functions f(x)=x+2 and g(x)=x^2, determine all values of x for which f(g(x))=g(f(x))

Answer 1

Hi Amber,

 

So first, you want to solve for each f(g(x)) and g(f(x)) separately.

 

f(g(x))           // Always solve for the inside function first. 

=f(x^2)         // If you plug in x into g(x) you get x^2 (just like if you plug in 3 for x you would get 9); you just replace the old function, g(x), with what it now equals, x^2.

=x^2+2        // If you plug x^2 into x for the function f(x) you get x^2+2

 

g(f(x))        

=g(x+2)        // When you plug in x into f(x)= you get x+2 (just like if you plug in 3 for x you would get 5); you just replace the old function, f(x), with what it now equals, x+2

=(x+2)^2      // If you plug in x+2 into x for the function g(x) you get (x+2)^2

 

We can now substitute the above into f(g(x))=g(f(x)) to get x^2+2=(x+2)^2.

 

Then we will solve for x.

x^2+2=(x+2)^2

x^2+2=x^2+4x+4    // Foil

2=4x+4                    // Subtract x^2 from both sides

-2=4x                       // Subtract 4 from both sides

x=-1/2                     // Divide both sides by 4

 

Hope this helps.

 

Best,

Ekarin P.

Answer 2

First, you will want to find your new functions: f(g(x)) and g(f(x)).

f(g(x)) = f(x^2) so everytime you see x in your original function, replace it with x^2

f(x^2) = x^2+2

 

g(f(x)) = g(x+2) = (x+2)^2 = x^2 + 4x + 4

 

Now set the two new functions equal to each other to find x.

 

x^2 + 2 = x^2 + 4x + 4

4x = -2

x = -1/2

 

Lastly, plug in x to f(g(x)) and g(f(x)) to check your work.  Side question: why do you plug into f(g(x)) and not f(x)?