Question

If $y = f(x)$ be monotonically decreasing or decreasing function of x and M is the median of variable x , then the median of y is
A $f(M)$
B $\dfrac{M}{2}$
C ${f^{ - 1}}(M)$
D None of these

Answer 1

Hint: As it is given that the function is be monotonically decreasing or decreasing function so let us suppose that $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)................\left( {{x_n},{y_n}} \right)$ be the coordinate of the function which is satisfy the $y = f(x)$ hence if we consider function is increasing then ${x_1} < {x_2} < ..... < M < ....... < {x_n}$ so the median of variable y is the value of function at the median . Similarly of decreasing function

Complete step-by-step answer:
In this question we have to find out the median of y where $y = f(x)$ be monotonically decreasing or decreasing function of x and M is the median of variable x is given
So first let us consider that the function is monotonically decreasing and $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)................\left( {{x_n},{y_n}} \right)$ be the coordinate of the function which is satisfy the $y = f(x)$
As we consider that the function is monotonically decreasing then ${x_1} < {x_2} < ............ < {x_n}$
Then it is given that the M indicates the median of the variable x hence it is lie in the mid of this series ${x_1} < {x_2} < ............ < {x_n}$ mean that ,
${x_1} < {x_2} < ..... < M < ....... < {x_n}$
so for the median of variable y is the $y = f(M)$ because M is the median of variable x so the Median of variable y is $f(M)$
So now let us consider that the function is monotonically decreasing and $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)................\left( {{x_n},{y_n}} \right)$ be the coordinate of the function which is satisfy the $y = f(x)$
As we consider that the function is monotonically decreasing then ${x_1} > {x_2} > ............ > {x_n}$
Then it is given that the M indicates the median of the variable x hence it is lie in the mid of this series ${x_1} > {x_2} > ............ > {x_n}$ mean that ,
${x_1} > {x_2} > ..... > M > ....... > {x_n}$
so for the median of variable y is the $y = f(M)$ because M is the median of variable x so the Median of variable y is $f(M)$
Hence from both increasing or decreasing the median of variable y is $f(M)$
So the option A is correct .

Note: A monotonically increasing function is one that increases as x does for all real x. A monotonically decreasing function, on the other hand, is one that decreases as x increases for all real x. In particular, these concepts are helpful when studying exponential and logarithmic functions.