# Why does f = g in implicit equations?

• B
• etotheipi
In summary, the conversation discusses an implicit equation of the form ##g(y)=f(x)## and the validity of the statement ##f=g##. It is clarified that the two functions are not necessarily identical and may have different domains. The final remark is deemed irrelevant and can be ignored without affecting the truth of the statement.
etotheipi
TL;DR Summary
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I saw something in my notes that I didn't understand... we have ##y=f(x)##, and consider an implicit equation of the form ##g(y) = f(x)##. They then say that ##f=g##. Why is that true? I would have thought$$f = \{ (a,f(a)) : a\in \mathbb{R} \} \subseteq \mathbb{R}^2$$whilst ##g## is just$$g = \{ (a,a) : a\in \mathbb{R} \} \subseteq \mathbb{R}^2$$What am I missing? Thanks!

I can't make much sense of that. Explicitly ##g(y) = y##.

etotheipi
PeroK said:
I can't make much sense of that. Explicitly ##g(y) = y##.

Thanks, I presumed it was a mistake of some variety but wanted to check!

g and f are not identical. They do not necessarily even have the same domain. If f(x) == 1, then all we know is that g(1)=1. g is undefined for all other input values.

Can you post an image of this section in your notes or type out exactly what it says? There might be hidden assumptions.

etotheipi
It's essentially:
"It is also possible to differentiate an expression of the form ##g(y) = f(x)## with respect to ##x##,$$\frac{dg(y)}{dy} \frac{dy}{dx} = \frac{df}{dx}$$Rearranging, we find that the implicit function ##g(y) = f(x)## gives$$\frac{dy}{dx} = \frac{df}{dx}/\frac{dg}{dy}$$Since the function is defined with ##f=g##, this looks like we are simply computing the quotient of two quotients."
The implicit differentiation is fine, but that final remark caught me a little off guard...

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etotheipi said:
It's essentially:The implicit differentiation is fine, but that final remark caught me a little off guard...

Yeah, that final remark looks weird. I would just ignore it.

etotheipi
etotheipi said:
It's essentially:The implicit differentiation is fine, but that final remark caught me a little off guard...
The last remark doesn't make a lot of sense. If we invert the latter derivative we have:
$$\frac{dy}{dx} = \frac{df}{dx}\frac{dy}{dg}$$ which looks like the chain rule, if we allow the association ##f \leftrightarrow g##.

I.e. if we imagine parametrising a curve ##x(t), y(t)##, then ##f## and ##g## have the same value on the curve:
$$\hat g(t) \equiv g(y(t)) = f(x(t)) \equiv \hat f(t)$$ and ##\hat f = \hat g##.

etotheipi
I agree you can ignore it the rest of the statement is true anyway even if you just delete the f=g part?

etotheipi

## 1. Why is f equal to g in implicit equations?

The equality of f and g in implicit equations is due to the definition of implicit equations. In an implicit equation, both f and g represent the same function, but are written in different forms. Therefore, they are interchangeable and can be considered equal.

## 2. Can f and g be different functions in implicit equations?

No, f and g must represent the same function in implicit equations. If they are different functions, they would not be interchangeable and the equation would not hold true.

## 3. How does f = g simplify implicit equations?

By setting f and g equal to each other, we can eliminate one of the variables and reduce the complexity of the equation. This makes it easier to solve for the remaining variable.

## 4. Are there any exceptions to f = g in implicit equations?

Yes, there are some cases where f and g may not be equal in implicit equations. This can happen when there are restrictions on the domain or range of the function, or when the equation is not defined for certain values of the variables.

## 5. Can f and g be expressed in any form in implicit equations?

Yes, f and g can be expressed in any form in implicit equations as long as they represent the same function. This includes polynomial, exponential, logarithmic, and trigonometric forms, among others.

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