Articles of inverse function

Composition of functions question

Am restricting this question to the elementary context of Riemann integrals and continuous functions $f,g.$ Because this came up in the context of another question, I would prefer to keep the examples from that question, at the risk of artificiality. Let $g:[0,4]\to \mathbb{R}$ such that $$\int_0^1g(x)dx = \int_3^4 4-g(x)dx.~~~~~(1)$$ An example would be $g(x) = […]

Finding the inverse of $f(x) = x^3 + x$

How can one find the inverse of functions like $f(x) = x^3 + x$? I know how to do it for explicit quadratic functions; how do I express $x$ as a function of $y$ here?

If the graphs of $f(x)$ and $f^{-1}(x)$ intersect at an odd number of points, is at least one point on the line $y=x$?

I was reading about intersection points of $f(x)$ and $f^{-1}(x)$ in this site. (Proof: if the graphs of $y=f(x)$ and $y=f^{-1}(x)$ intersect, they do so on the line $y=x$) Then, I saw this statement was wrote by N. S.: “If the graphs of $f(x)$ and $f^{-1}(x)$ intersect at a single point, then that point lies […]

Inversion of Trigonometric Equations

I’ve been playing around with finding the domain-restricted inverses of trigonometric equations using the inverse trigonometric equations. One of the easier formulas I came up with was the formula for the inverse $$a\cos^2x+b\sin^2x$$ The process I used to invert this was to use the pythagorean identities to turn it into a single trigonometric function: $$=a(\cos^2x+\sin^2x)+(b-a)\sin^2x$$ […]

Does $f\colon x\mapsto 2x+3$ mean the same thing as $f(x)=2x+3$?

In my textbook there is a question like below: If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$ As a multiple choice question, it allows for the answers: A. $11$ B. $5$ C. $\frac{1}{11}$ D. $9$ If what I think is correct and I read the equation as: $$f(x)=2x-3$$ then, $$y=2x-3$$ $$x=2y-3$$ $$x+3=2y$$ $$\frac {x+3} {2} […]

One-Way Inverse

My Algebra $2$ teacher stressed the fact that when you find the inverse $g$ of a function $f$, you must not only check that $$f \circ g=\operatorname{id}$$ but you must also check that $$g \circ f=\operatorname{id}$$ For example, if $$f(x)=x^2$$ then $$g(x)=\sqrt{x}$$ is not its inverse, because $$f(g(x))=\sqrt{x^2}=|x|\ne x$$ However, I feel that this is […]

Can there be an injective function whose derivative is equivalent to its inverse function?

Let’s say $f:D\to R$ is an injective function on some domain where it is also differentiable. For a real function, i.e. $D\subset\mathbb R, R\subset\mathbb R$, is it possible that $f'(x)\equiv f^{-1}(x)$? Intuitively speaking, I suspect that this is not possible, but I can’t provide a reasonable proof since I know very little nothing about functional […]

Derivation of inverse sine, what is wrong with this reasoning?

I’m trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don’t understand why the following reasoning leads to a wrong answer. Because $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$ If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$ Since $\sin(x) […]

How is the Inverse function theorem used to prove that the formulae in this question are the same?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x (x)=\rho_\alpha(\alpha)\left|\frac{\mathrm{d}x}{\mathrm{d}\alpha}\right|^{-1}\tag{A}$$ is the same formula as $$\rho_y(y)=\rho_x(\phi^{-1}(y))\left|\frac{\mathrm{d}\phi^{-1}}{\mathrm{d}y}\right|\tag{B}$$ Where $(\mathrm{A})$ and $(\mathrm{B})$ are formulae that relate the probability densities for random variables $x$, $\alpha$ in $(\mathrm{A})$ and $y$ and $x$ in $(\mathrm{B})$. It’s probably pretty simple to […]

$(\sin^{-1} x)+ (\cos^{-1} x)^3$

How do I find the least and maximum value of $(\sin^{-1} x)+ (\cos^{-1} x)^3$ ? I have tried the formula $(a+b)^3=a^3 + b^3 +3ab(a+b)$ , but seem to reach nowhere near ?