Difference between revisions of "1981 AHSME Problems/Problem 17"
(Created page with "The function <math>f</math> is not defined for <math>x=0</math>, but, for all non-zero real numbers <math>x</math>, <math>f(x)+f\left(\dfrac{1}x\right)=x</math>. The equation ...") |
m (→Problem) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
The function <math>f</math> is not defined for <math>x=0</math>, but, for all non-zero real numbers <math>x</math>, <math>f(x)+f\left(\dfrac{1}x\right)=x</math>. The equation <math>f(x)=f(-x)</math> is satisfied by | The function <math>f</math> is not defined for <math>x=0</math>, but, for all non-zero real numbers <math>x</math>, <math>f(x)+f\left(\dfrac{1}x\right)=x</math>. The equation <math>f(x)=f(-x)</math> is satisfied by | ||
| − | <math>\textbf{(A)}\ \text{exactly one real number} | + | <math>\textbf{(A)}\ \text{exactly one real number} \qquad \textbf{(B)}\ \text{exactly two real numbers} \qquad\textbf{(C)}\ \text{no real numbers}\qquad \\ \textbf{(D)}\ \text{infinitely many, but not all, non-zero real numbers} \qquad\textbf{(E)}\ \text{all non-zero real numbers}</math> |
| − | |||
| − | \qquad \textbf{(B)}\ \text{exactly two real numbers} | ||
| − | |||
| − | \qquad\textbf{(C)}\ \text{no real numbers}\qquad \\ | ||
| − | |||
| − | \textbf{(D)}\ \text{infinitely many, but not all, non-zero real numbers} | ||
| − | |||
| − | \qquad\textbf{(E)}\ \text{all non-zero real numbers}</math> | ||
==Solution== | ==Solution== | ||
Revision as of 20:30, 17 June 2021
Problem
The function 
is not defined for 
, but, for all non-zero real numbers 
, 
. The equation 
is satisfied by
Solution
Substitute with 
.
