Difference between revisions of "Euclid 2020/Problem 6"
| Line 1: | Line 1: | ||
(a) Suppose that the function <math>g</math> satisfies <math>g(x) = 2x - 4</math> for all real numbers <math>x</math> and | (a) Suppose that the function <math>g</math> satisfies <math>g(x) = 2x - 4</math> for all real numbers <math>x</math> and | ||
that <math>g^-1</math> is the inverse function of <math>g</math>. Suppose that the function <math>f</math> satisfies | that <math>g^-1</math> is the inverse function of <math>g</math>. Suppose that the function <math>f</math> satisfies | ||
| − | <math>g(f(g^-1(x))) | + | <math>g(f(g^-1(x))) = 2x^2 + 16x + 26</math> for all real numbers <math>x</math>. What is the value of |
| − | < | + | <math>f(\pi)</math>? |
| − | (b) Determine all pairs of angles < | + | (b) Determine all pairs of angles <math>(x; y)</math> with <math>0 \le x < 180</math> and <math>0 \le y < 180 that |
satisfy the following system of equations: | satisfy the following system of equations: | ||
</math>log_2(sin x*cos y)<math> = -3/2</math> | </math>log_2(sin x*cos y)<math> = -3/2</math> | ||
<math>log_2(sin x/cosy)</math> = 1/2$ | <math>log_2(sin x/cosy)</math> = 1/2$ | ||
Latest revision as of 04:30, 31 March 2023
(a) Suppose that the function 
satisfies 
for all real numbers 
and
that 
is the inverse function of . Suppose that the function 
satisfies
for all real numbers . What is the value of
?
(b) Determine all pairs of angles 
with 
and $0 \le y < 180 that
satisfy the following system of equations:$ (Error compiling LaTeX. Unknown error_msg)log_2(sin x*cos y)
= 1/2$
