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Questions  >>  CBSE XII  >>  Math  >>  Relations and Functions

Let the function $f\;:\;R \rightarrow R$ be defined by$ f(x)=\cos x,\forall \;x\;\in R$. What kind of a function is f?

1 Answer

  • A function $f: A \rightarrow B$ 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 $f(x)=\cos x ,\forall x,\in R$
Let $f(x)=f(y)=1$
Step1: Injective or One-One function:
but $0 \neq 2\pi$
Hence f is not a one-one function
Step 2: Surjective or On-to function:
Consider $y=2 \in R$
There does not exist any $x \in R$ such that $f(x)=\cos x=y=2$
$\cos x$ can take values from -1 to 1 only
Hence f is not a onto function
$\textbf{Solution: Therefore f is neither one-one nor onto}$
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