[[1]]is the domain and range of the following real function: f(x) = −|x|.

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Detailed Solution

f(x) = −|x|
As we know that
|x| = x,   x > 0
    = −x, x < 0
    = 0,   x = 0
Here, we are given a real function.
So, domain and range should be real numbers
And from above we clearly see that there is no value of ‘x’ which makes the function f(x) undefined.
Hence, we can say that the domain of the function is all real numbers.
Therefore, for given function f(x)= −|x|, Df:[R](all real numbers)
Also, from above the given function is negative of mode. Hence as we know that mode of ‘x’ is always positive but as there is negative sign outside. Which implies the final result will always be negative.
i.e.
|x| > 0 for all value of ‘x’
Then, -|x| will be negative for all values of ‘x’.
Therefore the range of the given function will always be less than zero.
Hence, range of f(x) is given as Rf:(−∞,0]
Therefore, from above we see that domain and range of given function f(x) = −|x| are R and (−∞,0] respectively.

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