PEA 2022 PEA Q30 | ISI MSQE

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2022 PEA Q30

$f : \mathbb R \to \mathbb R$ , $f(x) = (x^2 + 1)^{2022}$ .

Reveal answer and solution

Answer

Solution

1
$f(-x) = ((-x)^2 + 1)^{2022} = f(x)$ , so $f$ is even and hence not one-one (e.g.\ $f(1)=f(-1)$ ).
2
Also, $x^2 + 1 \geq 1$ , so $f(x) \geq 1$ for all $x$ , hence $f$ is not surjective onto $\mathbb R$ .

Answer structure / marking notes

Even functions on $\mathbb R$ cannot be injective unless restricted to $[0,\infty)$ .

Content note

Imported from public/resources/isi/msqe/solutions/pea/2022/ISI_MSQE_PEA_2022_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.