Relations & Functions
Greatest Integer Function
Grade 12
Question:
<p>If <math>f(x) = \sin\left(\frac{[x]^8}{x^2 + x + 1}\right)</math>, where <math>[ ]</math> denotes the greatest integer function, then</p>
<p>(a) <math>f</math> is one-one</p>
<p>(b) <math>f</math> is not one-one and non-constant</p>
<p>(c) <math>f</math> is constant function</p>
<p>(d) <math>f</math> is zero function</p>
Step-by-Step Solution
Key Concept: When the argument of sine becomes 0, sine evaluates to 0. Check if the argument is always 0 or reduces to 0.
<p>For all <math>x</math>, <math>[x]^8 \geq 0</math> since <math>[x]</math> is an integer.</p><p>Also, <math>x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} > 0</math> for all <math>x</math>.</p><p>Since <math>0 \leq \frac{[x]^8}{x^2 + x + 1}</math> and for integer values of <math>[x]</math>, when <math>[x] = 0</math>, we have <math>\frac{[x]^8}{x^2 + x + 1} = 0</math>.</p><p>Thus <math>\sin\left(\frac{[x]^8}{x^2 + x + 1}\right) = \sin(0) = 0</math>.</p><p>Therefore, <math>f(x) = 0</math>, so <math>f</math> is both a constant function and a zero function.</p>
Correct Answer: c, d