Q52. - Algebra 2

Question

For which function does is the range$\left\{y|y\le 0\right\}?$

$\begin{array}{l}\left(A\right)f\left(x\right)=-x\\ \left(B\right)f\left(x\right)=\left[\text{}\left[x\right]\text{}\right]\\ \left(C\right)f\left(x\right)=\left|x\right|\\ \left(D\right)f\left(x\right)=-\left|x\right|\end{array}$

Step-by-Step Solution

Answer

Option (D) $f\left(x\right)=-\left|x\right|$is correct

Step-1 –– Domain and range of the function.

The domain of the function is the set of all input values and range of the function is the set of all output 

Step-2 – Determine the given value

It is given that the Range is $\left\{y|y\le 0\right\}$

Step -3 – Solve for given range.

Consider the function, (A) $f\left(x\right)=y$

The domain and range of the function is all real numbers.

 Consider the function, (B) $f\left(x\right)=\left[\right[x\left]\right]$

The domain of the function is all real numbers and range of the function is all integer value. Because the function is greatest integer function.

 Consider the function, (C) $f\left(x\right)=\left|x\right|$

The domain of the function is all real numbers and range of the function is all real number greater than or equal to 0. 

Because the function is absolute value function therefore it will always produce a positive value or 0.

(D) Consider, the function

$f\left(x\right)=-\left|x\right|$

We know that, $f\left(x\right)=y$

Substitute the value$x=2$, then the value of y is as follows

$\begin{array}{l}y=-\left|x\right|\\ y=-\left|2\right|\\ y=-2\end{array}$

Hence, the value of $y\le 0$

The Range$\left\{y|y\le 0\right\}$ is true