Solve this equation?

Questions
,
1

The attached Notebook tells everything more clearly than I can here.

function2[x_] := Abs[3 + I x]

I’m failing to get Mathematica to solve the equation

function2[x] == 5

If I tell Mathematica the correct answer, it confirms I’m correct,

In[6]:= function2[4]
Out[6]= 5

but I was hoping there’s a way to get Mathematica to solve the equation like I did?

2

From Documentations:

$Assumptions
is the default setting for the Assumptions option used in such functions as Simplify, Refine, and Integrate.

So, it must be specified explicitly that x is positive in function Solve[]:

Solve[function1[x] == 5 && x > 0, x]

What does it mean function2 is equivalent to function1?

In[9]:= Solve[function2[x] == 5 && x > 0, x]
Out[9]= {{x -> 4}}
3

So, it must be specified explicitly that x is positive in function Solve[]:

Thank you very much.

What is it mean function2 is equivalent to function1?

What I meant by that is, when x is Real and x>0, function1[x]==function2[x]. On 2nd thought, I realize that’s not exactly true, because function2 yields a single positive result, while function1 yields one negative and one positive result. (I might still be wrong; I’m not as good at math as most people here.)