Hello all,

Determine the functions \(\displaystyle f:\mathbb R \rightarrow \mathbb R\) that verify the relationship \(\displaystyle f(x)=f^2(\{x\})-f([x])+1\) , \(\displaystyle \forall x\in \mathbb R\) and where \(\displaystyle \{x\}\) and \(\displaystyle [x]\) are defined as in the following site:

Home Math Notes Algebra II TRIGONOMETRY

INTEGER PART OF NUMBERS. FRACTIONAL PART OF NUMBER

Suppose, \(\displaystyle x\) is the real number.

Its integral part is the greatest integral number, that isnâ€²t exceed \(\displaystyle x\).

The integral part of number \(\displaystyle x\) denotes as \(\displaystyle [x]\).

The fractional number of number \(\displaystyle x\) is difference between the number and its integral part, i.e. \(\displaystyle xâˆ’[x]\) .

The fractional part of number \(\displaystyle x\) denotes as \(\displaystyle \{x\}\).

For example,

\(\displaystyle [3.47]=3;\{3.47\}=0.47;\)

\(\displaystyle [âˆ’2.3]=âˆ’3;\{âˆ’2.3\}=âˆ’2.3âˆ’(âˆ’3)=0.7;\)

\(\displaystyle [15]=15;\{15\}=0\)

All the best,

Integrator

Determine the functions \(\displaystyle f:\mathbb R \rightarrow \mathbb R\) that verify the relationship \(\displaystyle f(x)=f^2(\{x\})-f([x])+1\) , \(\displaystyle \forall x\in \mathbb R\) and where \(\displaystyle \{x\}\) and \(\displaystyle [x]\) are defined as in the following site:

Home Math Notes Algebra II TRIGONOMETRY

INTEGER PART OF NUMBERS. FRACTIONAL PART OF NUMBER

Suppose, \(\displaystyle x\) is the real number.

Its integral part is the greatest integral number, that isnâ€²t exceed \(\displaystyle x\).

The integral part of number \(\displaystyle x\) denotes as \(\displaystyle [x]\).

The fractional number of number \(\displaystyle x\) is difference between the number and its integral part, i.e. \(\displaystyle xâˆ’[x]\) .

The fractional part of number \(\displaystyle x\) denotes as \(\displaystyle \{x\}\).

For example,

\(\displaystyle [3.47]=3;\{3.47\}=0.47;\)

\(\displaystyle [âˆ’2.3]=âˆ’3;\{âˆ’2.3\}=âˆ’2.3âˆ’(âˆ’3)=0.7;\)

\(\displaystyle [15]=15;\{15\}=0\)

All the best,

Integrator

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