MHB Applying function to entire side of equation, not just terms

  • Thread starter Thread starter find_the_fun
  • Start date Start date
  • Tags Tags
    Function Terms
AI Thread Summary
When applying a function to an equation, the function must be applied to the entire side of the equation, not just individual terms. For example, in the equation ln|y| = ln|x| + C, it is correct to rewrite it as e^(ln|y|) = e^(ln|x| + C), but incorrect to separate the terms as e^(ln|y|) = e^(ln|x|) + e^C. Similarly, for sin{x} = x + y, it can be rewritten as x = arcsin(x + y), but not as x = arcsin(x) + arcsin(y). This principle ensures that the equality is maintained when applying functions. Misapplying this can lead to incorrect conclusions in mathematical reasoning.
find_the_fun
Messages
147
Reaction score
0
I hate to ask this but whenever applying a function to the equation, the arguments is the entire one side of the equation right?

What I mean is
$$
ln|y|=ln|x|+C$$

can be rewritten as $$e^{ln|y|}=e^{ln|x|+C}$$
but not $$e^{ln|y|}=e^{ln|x|}+e^C$$ ?

So the entire RHS or LHS becomes the argument?

Similarly $$\sin{x}=x+y$$
can be rewritten as $$x=\arcsin{(x+y)}$$
but not $$x=\arcsin{x}+\arcsin{y}$$

I keep messing this up and it's really annoying.
 
Mathematics news on Phys.org
Yes, it applies to the entirety of both sides, and the equality will hold for any function applied to both sides. Anything else is not guaranteed to hold for all functions.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top