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RELATIONS AND FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRELATIONS AND FUNCTIONS4 Marks
Question
The relation f is defined by $\text{f(x)}=\begin{cases}\text{x}^2,0\leq\text{x}\leq3\\3\text{x},3\leq\text{x}\leq10\end{cases}$ The relation g is defined by $\text{g(x)}=\begin{cases}\text{x}^2,0\leq\text{x}\leq2\\3\text{x},2\leq\text{x}\leq10\end{cases}$ Show that f is a function and g is not a function.
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Answer

It is observed that for $0\leq\text{x}<3,\text{f(x)}=\text{x}^2$ $3<\text{x}\leq10,\text{f(x)}=3\text{x}$ Also, at $\text{x}=3,\text{f(x)}=3^2=9$ or $\text{f(x)}=3\times3=9$ i.e., at $\text{x}=3,\text{f(x)}=9$
Therefore, for $0\leq\text{x}\leq10,$ the images of f(x) are unique. Thus, the given relation is a function.
The relation g is defined as, $\text{g(x)}=\begin{cases}\text{x}^2,0\leq\text{x}\leq2\\3\text{x},2\leq\text{x}\leq10\end{cases}$ It can be observed that for $x = 2, g(x) = 2^2= 4$ and $g(x) = 3 × 2 = 6$
Hence, element 2 of the domain of the relation g corresponds to two different images i.e., 4 and 6. Hence, this relation is not a function.

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