Consider the real-valued function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=ax+b$. Assume that $a,b,c,d\in\mathbb{R}$. The function $f$ is bijective and is invertible (it also satisfies the horizontal/vertical tests). Thus, $f$ has an inverse function.
First, we write $y=ax+b$.
Now, suppose the inverse is $x=cy+d$.
Next, substitute $x=cy+d$ into $y=ax+b$ yields $$x=c(ax+b)+d=acx+bc+d.$$
Then, we rewrite the above equation and so $x(ac-1)+bc+d=0$.
Hence, we have $ac-1=0$, so $c=1/a$ and $bc+d=0$, so $-bc=0$ or $d=-b/a$.
In conclusion, the inverse function $f^{-1}(x)=\frac{1}{a}x-\frac{b}{a}$.
We can verify like $f^{-1}(f(x))=\frac{1}{a}(ax+b)-\frac{b}{a}=x$ for all $x$.
Are there any issues with what I have written above? Is the proof rigorous? Do I abuse any math notion?
Would the above method work for finding the inverse function of a cubic polynomial function, say $f(x)=y=ax^3+bx^2+cx+d$? For example, assuming that $x=\alpha y^3+\beta y^2+\gamma y+\delta$ is the inverse and substituting into $f(x)$?
I'd appreciate any help. Thank you.
In the linear case, the writing and math are both good. Note that how much should be included depends upon your goal as a writer. If your goal is to demonstrate not only what the answer is but how you arrived at that answer, then everything you've written is great to keep. However, if your goal is to give the answer, then everything from "First, we write ..." to "... In conclusion" can be omitted: the statement of what the inverse function is, and the verification that it is the inverse function, is all that is needed.
The cubic case will be substantially more difficult, in part because not all cubic functions are invertible: consider $f(x) = x^3 - x$. Even for a cubic function that is invertible, such as $g(x) = x^3 + x$, the formula for the inverse will be complicated: just solving $g(x) = 1$ by itself is already quite complicated. And these cubics don't even have quadratic terms yet....