A) $f$ is injective and surjective

B) $f$ is injective only

C) $f$ is surjective only

D) $f$ is neither injective nor surjective

Note: This is the 5th part of a 5 part question, which is split as 5 separate questions here.

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B) $f$ is injective only

C) $f$ is surjective only

D) $f$ is neither injective nor surjective

Note: This is the 5th part of a 5 part question, which is split as 5 separate questions here.

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- A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function.
- A function$ f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.

Given $f:Z \rightarrow Z$ defined by $f(x)=x^3$.

Let $x$ and $y$ be two elements in $Z$.

Step1: Injective or One-One function:

For an injective or one-one function, $f(x) = f(y)$

$ \Rightarrow x^3 = y^3$$ \Rightarrow x = y.$

Therefore $f:Z \rightarrow Z$ defined by $f(x)=x^3$ is one-one or injective.

Step 2: Surjective or On-to function:

For an on-to function, for every $y \in Y$, there exists an element x in X such that $f(x) = y$.

$ \Rightarrow$ For every $y \in Z$ there must exist $ x$ such that $ f(x)=x^3 = y$.

However, we see that for $y=2$, there is no $x \in Z$ such that $f(x) = x^3 = 2$.

Therefore $f:Z \rightarrow Z$ defined by $f(x)=x^3$ is not onto or surjective.

Solution: $f:Z \rightarrow Z$ defined by $f(x)=x^3$ is one-one or injective but not onto or surjective.

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