No. of solutions

Recently this problem was given to me by a friend. I am unable to understand how to approach this. $\textbf{NOTE}$ : I do not know about the validity of the problem.

If $f(x)=sinx$ has exactly $2$ solutions, how many solutions can $f(\frac{1}{x})=sinx$ have? $[f(x):\mathbb{R}\to\mathbb{R}]$

The answer given is $\boxed{3}$

#Calculus #Functions #Solutions

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Given only the information provided, my answer would be "arbitrarily many". Are you missing out some conditions?

E.g. Is the function defined from the reals to the reals? Must the function be continuous? differentiable? strictly increasing? etc.

Calvin Lin Staff - 6 years ago

yes, i am sorry, the function is $\text{ is continuous and differentiable}$ and is defined from $:\mathbb{R}\to\mathbb{R}$ again, i am extremely sorry for any confusion over this subject

Aritra Jana - 6 years ago

Even under those assumptions, my answer is still "arbitrarily many".

Think about what happens if $f(x) = 0$ in a neighborhood about $0$ .

Calvin Lin Staff - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$