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Question
Which of the following functions form Z to itself are bijections?
Options
\[f\left( x \right) = x^3\]
\[f\left( x \right) = x + 2\]
\[f\left( x \right) = 2x + 1\]
\[f\left( x \right) = x^2 + x\]
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Solution
f is not onto because for y = 3∈Co-domain(Z), there is no value of x∈Domain(Z)
\[ x^3 = 3\]
\[ \Rightarrow x = \sqrt[3]{3} \not\in Z\]
⇒ f is not onto
So, f is not a bijection
(b) Injectivity:
Let x and y be two elements of the domain (Z), such that
\[x + 2 = y + 2\]
\[ \Rightarrow x = y\]
Surjectivity:
Let y be an element in the co-domain (Z), such that
\[ \Rightarrow y = x + 2\]
\[ \Rightarrow x = y - 2 \in Z \left( Domain \right)\]
So, f is a bijection.
\[ \Rightarrow 4 = 2x + 1\]
\[ \Rightarrow 2x = 3\]
\[ \Rightarrow x = \frac{3}{2} \not\in Z\]
So,f is not a bijection.
\[\]
\[\left( d \right) f\left( 0 \right) = 0^2 + 0 = 0\]
\[and f\left( - 1 \right) = \left( - 1 \right)^2 + \left( - 1 \right) = 1 - 1 = 0\]
⇒ 0 and -1 have the same image.
⇒ f is not one-one.
So,fis not a bijection.
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