<p>Which of the following functions from Z into Z are bijections?</p>
<p style="display:inline">f (x) = x<sup>3</sup></p>
<p style="display:inline">f (x) = 2x + 1</p>
<p style="display:inline">f (x) = x<sup>2</sup> + 1</p>
<p style="display:inline">f (x) = x + 2</p>
Step-by-Step Solution
Key Concept: A function is a bijection if it is both injective, where $f(x)=f(y)$ implies $x=y$, and surjective, where every element in the codomain has a pre-image in the domain.
<p>Injectivity: Let x, y <span class="math-tex">$ \in $</span> Z, then, f(x) = f(y) <span class="math-tex">$ \Rightarrow $</span> x + 2 = y + 2 <span class="math-tex">$ \Rightarrow $</span>x = y.<span class="math-tex">$ \Rightarrow $</span>f is one-one.<br />
Surjectivity: Let f(x) = y , where y <span class="math-tex">$ \in $</span> Z, .<span class="math-tex">$ \Rightarrow $</span> x + 2 = y .<span class="math-tex">$ \Rightarrow $</span> x = y – 2 <span class="math-tex">$ \in $</span> Z, <span class="math-tex">$ \Rightarrow $</span> f is onto.<br />
Therefore, f is a bijective function.</p>
Correct Answer: D