Sets, Relations & Functions
General
Grade 11
Question:
<p>Find the inverse of <span class="math-inline">\(f(x)=\dfrac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}}\)</span>, <span class="math-inline">\(x\in(-1,1)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Let u=8^{2x}. y=(u²-1)/(u²+1). Solve: u²=(1+y)/(1-y). So 4x=log₈((1+y)/(1-y))=(log₈e)ln((1+y)/(1-y)). f⁻¹(x)=(log₈e/4)ln((1+x)/(1-x)).</p><div class="trap-box"><strong>Trap:</strong> Remember log₈A=(log₈e)ln A.</div><div class="key-concept"><strong>Key Concept:</strong> (t-1/t)/(t+1/t) = (t²-1)/(t²+1) standard form</div></div>
Correct Answer: f⁻¹(x) = (log₈e/4)·ln((1+x)/(1-x))