If \(x=\log_4\left(\dfrac{2f(x)}{1-f(x)}\right)\), then find \(f(2010)+f(-2009)\)
1
From \(x=\log_4\left(\dfrac{2f(x)}{1-f(x)}\right)\), raising both sides as powers of 4: \(4^x=\dfrac{2f(x)}{1-f(x)}\).
Solving for f(x): \(4^x(1-f(x))=2f(x)\Rightarrow 4^x=f(x)(2+4^x)\Rightarrow f(x)=\dfrac{4^x}{2+4^x}\).
Now compute \(f(1-x)=\dfrac{4^{1-x}}{2+4^{1-x}}\). Multiply numerator and denominator by \(4^x\): \(f(1-x)=\dfrac{4}{2\cdot4^x+4}=\dfrac{2}{4^x+2}\).
Add: \(f(x)+f(1-x)=\dfrac{4^x}{4^x+2}+\dfrac{2}{4^x+2}=\dfrac{4^x+2}{4^x+2}=1\).
So for any x, \(f(x)+f(1-x)=1\). Since \(2010+(-2009)=1\), we have \(-2009=1-2010\), so \(f(-2009)=f(1-2010)\).
Therefore \(f(2010)+f(-2009)=f(2010)+f(1-2010)=1\).
Similar Questions
Let \(f:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\to R\) be given by \(f(x)=(\log(\sec x+\tan x))^3\), then which is not true:
Important Questions from Types of Functions
The number of one-to-one functions from {1, 2, 3} to {1, 2, 3, 4, 5} is
Consider the following statements:
1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.
2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.
Which of the above statements is/are correct?
For \(x = \frac{{\sqrt \pi }}{2}\) , what is the value of [ho(gof)](x)?
What is [fo(fof)](2) equal to?
If A = {1, 2, 3} and B = {4, 5, 6}, then which of the following is bijective function?
