Question

The function f is defined by
$f\left(x\right)=\{\begin{array}{c}{x}^2,0\le x\le 3\\ 3x,3\le x\le 10\end{array}$
The relation g is defined by
$g\left(x\right)=\{\begin{array}{c}{x}^2,0\le x\le 2\\ 3x,2\le x\le 10\end{array}$
Show that f is a function and g is not a function.

Answer 1

We have,
$f\left(x\right)=\{\begin{array}{c}{x}^2,0\le x\le 3\\ 3x,3\le x\le 10\end{array}$
and, $g\left(x\right)=\{\begin{array}{c}{x}^2,0\le x\le 2\\ 3x,2\le x\le 10\end{array}$
Now, $f\left(3\right)=(3{)}^2=9$ and $f\left(3\right)=3\times 3=9$
and $g\left(2\right)=(2{)}^2=4$ and $g\left(2\right)=3\times 2=6$
We observe that f(x) takes unique value at each point in its domain [0, 10]. However g(x) does not take unique value at each point in its domain [0, 10].
Hence, g(x) is not a function.