# Tag Info

### Is e^x is strictly increasing or monotonically increasing or both

Since $f'(x) = e^{x}$ and $e^{x}>0$ for all real $x$, it follows that $f(x)$ is a strictly increasing function on the real line. It also follows that $f'(x)\geq 0$, so based on your definition of a ...

• 134
1 vote

### Characterize a specific measurable space

$f+\frac 1 f \ge 2$ so integrability of $f$ and $\frac 1 f$ implies integrability of the constant $2$ which means $\mu (E) <\infty$. Conversely, if $\mu (E)<\infty$ then we can take $f\equiv 1$...
• 39.2k
Accepted

### Determine all possible values ​of the sum $S=a+b+c+d$

Continuing on your answer. Since, $f(f(a)) - f(b) = 2022 = c^2a - cb + cd$ $b-a=20$ $d-c=25$ (mistake in your answer) Expanding this equation further gives us: $c(ca-b+d) = 2022$ and on using the ...
Accepted

• 174k

### How to find the range of a quadratic function we can't use the quadratic formula?

Alternative approach: To analyze the behavior of $~f(x) = \dfrac{x}{1 + x^2},~$ start by examining its 1st derivative. Then, either also examine its 2nd derivative, or look for alternate approaches. ...
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### How to find the range of a quadratic function we can't use the quadratic formula?

According to the AM-GM inequality, one concludes that \begin{align*} 1 + x^{2} \geq 2\sqrt{x^{2}} = 2|x| & \Rightarrow \frac{|x|}{1 + x^{2}} \leq \frac{1}{2}\\\\ & \Rightarrow \left|\frac{x}{1 ...
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1 vote

### How to find the range of a quadratic function we can't use the quadratic formula?

You can only use the quadratic formula for $ax^2+bx+c=0$ when $a\neq0$. So, you need to consider separately the possibility $y=0$, and in that case you get $x=0$, consistent with $y=x/(1+x^2)$. Hope ...
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### How to find the range of a quadratic function we can't use the quadratic formula?

What you assumed in your question is wrong. You can't claim $y \neq 0$ because it is zero if $x=0$. Also, notice that $f(x)$ is a continuous function in $\Bbb R$. But to find the range, you need to ...
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1 vote

• 18.9k
1 vote
Accepted

• 144k

### Back to basics: Why do we care about symmetry of functions?

I think we can get some information from the symmetry of a function. e.g. a simple sin function, it has period and we can use it to solve equations. and we can only draw a period instead of a whole ...
1 vote
Accepted

### How to say these two distinct functions have the same structure?

It is possible to phrase this in very abstract terms along the lines I indicted in my comment but I think that would obscure the underlying idea. Here's a simple definition that I think captures the ...
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### Prove that assuming $f:S\rightarrow T$, $f$ is a bijection iff there is $g:T\rightarrow S$ such that $f\circ g$ and $g\circ f$ are identity maps

Reverse implication seems right, but we have a problem with the forward implication, let's discuss it. Since you have done the reverse implication right (or so we have assumed) let's just focus on the ...
1 vote

• 6,004
1 vote
Accepted

### Determining $t(x)$ from $\frac{dx}{dt}$?

Numerically, the calculation of $x(t)$ is a table with two rows, $t$, and $x$, and it simultaneously calculates $t(x)$: calculating $t(x)$ from is simply a matter of reading the table inversely. ...
1 vote

There seems to be no useful "next best thing" that is true! To see this, consider an integral domain (containing a multiplicative neutral element $1$) $R$, and assume that $x , y , u , ... • 6,766 1 vote Accepted ### Proving$f + c = O(f)$doesn't always hold- where is my mistake? Your proof is correct. To simplify it you can just give the counterexample you stated: Taking$f(x)=1/x$, and$c=1$, there is no$k$such that$f(x)+c < kf(x)$for all$x\$. The statement as given ...
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