Assignment -1
Grade Class 12
Question:
<p>Which of the following functions from<span class="math-tex">\(A=\{x:-1 \leq x \leq 1\}\)</span> to itself are bijections?</p>
<p style="display:inline"><span class="math-tex">\(f(x)=\frac{x}{2}\)</span></p>
<p style="display:inline">k(x) = x<sup>2</sup></p>
<p style="display:inline"><span class="math-tex">\(g(x)=\sin \left(\frac{\pi x}{2}\right)\)</span></p>
<p style="display:inline">h(x) = |x|</p>
Step-by-Step Solution
Key Concept: A function is a bijection if it is both one-one (each input has a unique output) and onto (the range must exactly match the codomain).
<p>Given that <span class="math-tex">\(A=\{x:-1 \leq x \leq 1\}\)</span><br />
<span class="math-tex">\(f(x)=\frac{x}{2}\)</span><br />
It is one-one but not onto.<br />
<span class="math-tex">\(g(x)=\sin \left(\frac{\pi x}{2}\right)\)</span><br />
It is bijective as it is one-one and onto with range [-1, 1].<br />
h(x) = |x|<br />
It is not one-one because h(-1) = 1 and h(1) = 1.<br />
k(x) = x<sup>2</sup><br />
It is not one-one because k(-1) = 1 and k(1) = 1.</p>
Correct Answer: C