Articles of inverse function

Let $f(x) = 6\operatorname{arcsec}(2x)$. Find $f'(x)$.

Let $f(x) = 6\operatorname{arcsec}(2x)$. Find $f'(x)$. $$6 \cdot \frac 1{2x\sqrt{(2x)^2-1}} \cdot 2$$ $$\frac{12}{2x\sqrt{4x^2-1}}$$ $$\frac 6 {x\sqrt{4x^2-1}}$$ Why is that wrong?

Knowing which theorem of calculus to use to prove number/nature of solutions

How does one go about showing that the equation $$ arctanx = x^2 $$ has at least one solution and then in turn show that the equation has exactly one positive solution? I figured for part a) “show at least one solution” it could be possible to rewrite the equation as $$ x^2 – arctanx […]

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} $$ Then why isn’t this true? $$ {\tan}^{-1}\theta = \dfrac{1}{{\tan}\theta} $$

Inverse Function Theorem and Injectivity

I have the following problem (which involved the Inverse Function Theorem and Injectivity): Let $f(x_1,x_2,x_3)=(u(x_1,x_2,x_3),v(x_1,x_2,x_3),w(x_1,x_2,x_3))$ be the mapping of $\mathbb{R}^3$ and $\mathbb{R}^3$ given by $u=x_1$, $v=x_1^2+x_2$, and $w=x_1+x_2^2+x_3^3$. The Jacobian of $f$ is given by $$ \begin{pmatrix} 1 & 0 & 0 \\ 2x_1 & 1 & 0 \\ 1 & 2x_2 & 3x_3^2 \end{pmatrix} […]

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 […]