Question

Let $f:[0,10]\to [1,20]$ be a function defined as
$f(x)=\{\begin{array}{ll}\frac{60-5x}{3},& 0\le x\le 6\\ 10,& 6\le x\le 7\\ 31-3x,& 7\le x\le 10\end{array}$
then $f$ is

• A. bijective function
• B. one-one but not on to function
• C. onto but not one-one function
• D. neither one -one nor onto function

Answer 1

Answer: (C)

We have$f(x)=\{\begin{array}{ll}\frac{60-5x}{3},& 0\le x\le 6\\ 10,& 6\le x\le 7\\ 31-3x,& 7\le x\le 10\end{array}$
Now, $f(x)=10\forall x\in [6,7]$
So, $f(x)$ is not one-one.
Again,
$0\le x\le 6$
$\Rightarrow -30\le -5x\le 0$
$\Rightarrow 60-30\le 60-5x\le 60$
$\Rightarrow 10\le \frac{60-5x}{3}\le 20$ and $7\le x\le 10$
$\Rightarrow -30\le -3x\le -21$
$\Rightarrow 1\le 31-3x\le 10$
$\therefore$ Range of $f(x)=[1,20]$
It is given that co-domain of $f(x)=[1,20]$
$\therefore$ Range of $f(x)=$ co-domain of $f(x)$
So, $f(x)$ is onto.