Sets, Relations & Functions
General
Grade 11
Question:
<p>Find range of <span class="math-inline">\(f(x)=\log_{\sqrt{5}}\left(3+\cos\left(\frac{\pi}{4}+x\right)+\cos\left(\frac{\pi}{4}-x\right)+\cos\left(\frac{3\pi}{4}+x\right)-\cos\left(\frac{3\pi}{4}-x\right)\right)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Simplify: cos(π/4+x)+cos(π/4-x)=2cos(π/4)cos x=√2 cos x. cos(3π/4+x)-cos(3π/4-x)=-√2 sin x. Inner = 3+√2(cos x-sin x) = 3+2cos(x+π/4). Range of inner: [1,5]. log_{√5}[1,5]=[0,2].</p><div class="trap-box"><strong>Trap:</strong> Don't find log range before simplifying trig expression.</div><div class="key-concept"><strong>Key Concept:</strong> Sum-to-product identities before logarithm range</div></div>
Correct Answer: [0, 2]