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| + | Find all functions <math>f: \mathbb{R} \to \mathbb{R}</math> such that<cmath>f(xonkrbo)=xonkrbo</cmath>for all real numbers <math>x,o,n,k,r,</math> and <math>b.</math> | ||
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| + | Let <math>A</math> be the set of positive real numbers. Determine, with proof, if there exists at least one function <math>f : A\to A</math> such that<cmath>f(x^x)=f(x)^{f(x)}</cmath>for all real <math>x</math> in <math>A.</math> | ||
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| + | Determine all functions <math>f:\mathbb{R} \to \mathbb{R}</math> such<cmath>f(x+y)=f(y)</cmath>for all real numbers <math>x</math> and <math>y.</math> | ||
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| + | Find all functions <math>f:\mathbb{R}\to \mathbb{R}</math> such that<cmath>f\left(x+\dfrac{1}{x}\right)+f\left(y+\dfrac{1}{y}\right)+f\left(z+\dfrac{1}{z}\right)=1</cmath>for all real numbers <math>x,y,z\neq 0.</math> | ||
Revision as of 00:52, 27 January 2024
test
1434
xonk
Find all functions 
such that![\[f(xonkrbo)=xonkrbo\]](http://latex.artofproblemsolving.com/f/c/6/fc675824e77e68b11f58e78a79f1b6b82990fe3f.png)
for all real numbers 
and 
Let 
be the set of positive real numbers. Determine, with proof, if there exists at least one function 
such that![\[f(x^x)=f(x)^{f(x)}\]](http://latex.artofproblemsolving.com/f/3/6/f36a23cab5d3ed8b6e2fd4df576cf1a4ff59c51a.png)
for all real 
in 
Determine all functions 
such![\[f(x+y)=f(y)\]](http://latex.artofproblemsolving.com/c/1/1/c1117c05e7ecb5262a9e39c01ff6e8e1d644324c.png)
for all real numbers and 
Find all functions 
such that![\[f\left(x+\dfrac{1}{x}\right)+f\left(y+\dfrac{1}{y}\right)+f\left(z+\dfrac{1}{z}\right)=1\]](http://latex.artofproblemsolving.com/a/6/1/a6163664953f7b976b02b87cab3274191aca527f.png)
for all real numbers 
