Question

Identify the function based on the description given below.

1. Its graph is symmetrical about y-axis

2. Domain is R and range is $[0,\infty )$

3. It gives distance from origin

4. Output of the function is always numerically equal to input

5. Its graph is linear and is inclined at an angle of ${45}^{\circ }$ for $x\ge 0$

• A.
• B.
• C. x
• D. |x|
• E. -x

Answer 1

Answer: (B), (D)

The correct options are
B

D

|x|

We will look at the options and eliminate the function which does not satisfy the relations given.First relation given is its graph is symmetrical about y-axis. $\sqrt{x}$ is not defined for values less than zero. So its graph can't be symmetrical about y-axis. Next option is $\sqrt{x^2}$.

This is same as |x|, because

$\sqrt{x^2}$= if $x\ge 0$

= -x if $x\le 0$

This is definition is same as the definition of |x| or modulus function. Both the functions are symmetric about y-axis.

The next option x is not symmetric about y-axis, same with -x.

Now, we are left with two options B and D. both of them are |x|.

Now, we just have to check if other conditions are satisfied by this function.

2. |x| is defined for all $xϵR$ and it's never negative

$\Rightarrow$ 2 is correct

3. This is the definition (alternate) of |x|.

4. This means if $\pm 2$ is the input it gives 2 as output. This is true for |x|.

5. If $x\ge 0$ |x| is equal to x. This is a linear function and it is inclined at an angle of ${45}^{\circ }$.

So |x| and $\sqrt{x^2}$ satisfy all the conditions given.