Question

The number of linear functions which map $\left[ -1,1 \right]$ to $\left[ 0,2 \right]$ are.
A)One
B)Two
C)Four
D)Three

Answer 1

Hint: A linear function must be increasing or decreasing.In an onto function range is equal to the codomain.It is given codomain=$\left[ 0,2 \right]$.And consider a linear function $f(x)=ax+b$ where $a\ne 0$ and whose range is $\left[ -\infty ,\infty \right]$ but in this question its end points that is minimum or maximum is from $\left[ -1,1 \right]$ which maps $\left[ 0,2 \right]$.

Complete step by step answer:
Let us consider a linear function $f(x)=ax+b$, which is linear
Since $f(x)=ax+b$is linear ,it should be either increasing or decreasing.
Let us consider,
Case(1): Let $f(x)$ be increasing function
That is$f\left( x \right)\ge 0$ it means $a\ge 0$
As it is given $\left[ -1,1 \right]$ maps on $\left[ 0,2 \right]$
Therefore, the function values is equals to $f(-1)=0$ and $f(1)=2$
Since $f(x)$ is onto, it is increasing towards $\left[ -1,1 \right]$⟶$\left[ 0,2 \right]$
Therefore, we get the equations as shown below
$-a+b=0$ and $a+b=2$
On solving above two equations we get a,b values as follows,
$a=1$ and $b=1$
Therefore function $f(x)$becomes as
$f(x)=x+1$⟶equation(1)
Case(2): Let $f(x)$be decreasing function,
That is $f(x)\le 0$
As it is given $\left[ -1,1 \right]$ maps on $\left[ 0,2 \right]$
Therefore the function values becomes ,$f(-1)=2$ and $f(1)=0$
Since $f(x)$onto ,decreasing in $\left[ -1,1 \right]$⟶$\left[ 0,2 \right]$
Now the resultant equations are $-a+b=2$ and $a+b=0$
On solving above two equations we the a,b values as follows,
$a=-1$ and $b=1$
Therefore, $f(x)=-x+1$⟶equation(2)
Hence, we obtain two linear functions that is equation(1) and equation(2)
$f(x)=x+1$ and $f(x)=-x+1$
Therefore ,the correct option is (B)

Note: Here the students must remember that for an onto function there exist an “x” for every “y”.And it is also called as surjective function. And a linear function is an increasing or decreasing function. According to that, the linear function must be solved. And choose the correct option from the given.