Question

Find the domain and range of the following real function:
(i) f(x) = –|x| (ii) f(x) = √(9 – x2) 
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Answer 1

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Find the domain and range of the following real function:

(i) f(x) = –|x| (ii) f(x) = √(9 – x2) 

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✴(i) Given,

➡️f(x) = –|x|, x ∈ R

➡️We know that,

➡️As f(x) is defined for x ∈ R, the domain of f is R.

➡️It is also seen that the range of f(x) = –|x| is all real numbers except positive real numbers.

➡️Therefore, the range of f is given by (–∞, 0].

✴(ii) f(x) = √(9 – x2)

➡️As √(9 – x2) is defined for all real numbers that are greater than or equal to –3 and less than or equal to 3, for 9 – x2 ≥ 0.

➡️So, the domain of f(x) is {x: –3 ≤ x ≤ 3} or [–3, 3].

➡️Now,

➡️For any value of x in the range [–3, 3], the value of f(x) will lie between 0 and 3.

➡️Therefore, the range of f(x) is {x: 0 ≤ x ≤ 3} or [0, 3].

Answer 2

Answer:

=> values of x will be restricted such that ( 9 - x^2 ) > 0

9 - x^2 = ( 3 + x ) ( 3 - x )

this is parabola of inverted U shape, whose value is +ve in between roots.

so ( 9 - x^2 ) > 0 for x in between -3 & 3

so domain is x = [ -3 , 3 ]

range = ( 0 , max of f(x) )

f(x) is maximum at mid point of roots, i.e. f max = f(0) = 3

so range = ( 0 , 3 )

Step-by-step explanation:

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