# Tagged Questions

Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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### Help with Inverse Function and Composition of Functions?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on ...
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### Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$

The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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### Show that f(x)=e^x from set of reals to set of reals is not invertible…

Yes, this is my question... How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, ...
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### Prove that if $f$ is increasing then so is $f^{-1}$

Prove that if $f$ is increasing then so is $f^{-1}$, when $f$ is a one-to-one function. I'm having trouble figuring out how to get started with this question. I'm assuming it has something to do with ...
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### Find inverse $z$-transform of $\frac{5}{z^{2}-z-6}$

How can I find inverse z transform of $$X(z)=\frac{5}{z^{2}-z-6}$$ What I did: first i factored denominator and i got (z+2)(z-3), now we get A(-2^{n}) + b(3^{n}). To get A and B i used Partial ...
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### Cofactor expansion to check if matrices is invertible.

I gave question regarding a co-factor expansion question. I understand that an easy way to check if a matrices is invertible is to do co-factor expansion and if $A \ne 0$ then its invertible. I'm ...
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### Find inverse $z$-transform of $\dfrac{(z-1)^2}{z^3}$

How can I find inverse z transform of $$X(z)=\frac{(z-1)^{2}}{z^{3}}$$ What I did: I am thinking to do Partial Fraction Decomposition or long division. Is there another method ?
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### Write $(h \circ f)^{-1}$ when $h(x)= x \ln(3 + x)$ and $f(x) = x^2 − x$

I have arrived up to a point but haven't solved it yet: $$(h \circ f)^{-1} = y= (x ^2 − x )· \ln(3 + x^ 2 − x)$$ $$x = (y^ 2 − y )\cdot \ln(3 + y^ 2 − y)$$ Any suggestions? Thank you
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### Determinant of $2\times 2$ matrix over $\mathbb{Z}/2\mathbb{Z}$

I have to prove that for any square matrix that is in $M_2(\mathbb{Z}/2\mathbb{Z})$ it is invertible if and only if its determinant is not $0$. Here are my thoughts: Since all entries are modulo $2$, ...
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### How to prove that $\,I-\gamma\, T\,\;\big(\,T\,$ is stochastic matrix, $\,0 \le \gamma \lt 1\,\big)\,$ is invertible

In case $\,T\,$ is right stochastic matrix (sum of each row is $1$) and $\,0 \le \gamma \lt 1$, Is there any way to prove that $\,I-\gamma \,T\,$ is invertible?
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### How compute Inverse Fourier of this function $\mathcal{F}^{-1} \Big\{\frac{1}{iw - \alpha^2 k^2}\Big\}$

I need to compute this Inverse Fourier Transform to arrive at the given result $$p(t) = \mathcal{F}^{-1} \Big\{\frac{1}{iw - k^2\alpha^2 }\Big\} = -\sqrt{2\pi}H(t)e^{-k^2\alpha^2t}$$ Where $H(t)$ ...
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### Find inverse of $7x^{2}-112x+448$

Given the function $\; f(x) = 7x^{2}-112x+448, \;$ for $x\ge 8, \;$ find $\displaystyle \;$ $f^{-1}(x)$. To find inverse, I should just solve for x in terms of y: $$y = 7x^{2}-112x+448$$ I can ...
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### another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
The formula for the Inverse Laplace Transform is (Bromwich Intergal): $$f_{(t)}=\frac{1}{2\pi i}\lim_{x\to\infty}\int_{\alpha-x i}^{\alpha+x i} \left(e^{st}\cdot F_{(s)}\right) \text{d}s$$ My ...
Help! I need to simplify this expression. I'm not even sure where to start. $$\tan{(\arccos{(\frac{x}{4})})}$$