Question

Find the range of each of the following functions.
(i) f(x) = 2 – 3x, x ∈ R, x > 0.
(ii) f(x) = x2 + 2, x is a real number.
(iii) f(x) = x, x is a real number.
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Answer 1

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$\Large\fbox{\color{purple}{QUESTION}}$

Find the range of each of the following functions.

(i) f(x) = 2 – 3x, x ∈ R, x > 0.

(ii) f(x) = x^2 + 2, x is a real number.

(iii) f(x) = x, x is a real number.

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$\bf\Huge\red{\mid{\overline{\underline{ ANSWER }}}\mid }$

✴(i) Given,✴

➡️f(x) = 2 – 3x, x ∈ R, x > 0.

➡️We have,

➡️x > 0

➡️So,

➡️3x > 0

➡️-3x < 0 [Multiplying by -1 both the sides, the inequality sign changes]

➡️2 – 3x < 2

➡️Therefore, the value of 2 – 3x is less than 2.

➡️Hence, Range = (–∞, 2)

✴(ii) Given,✴

➡️f(x) = x2 + 2, x is a real number

➡️We know that,

➡️x2 ≥ 0

➡️So,

➡️x2 + 2 ≥ 2 [Adding 2 both the sides]

➡️Therefore, the value of x2 + 2 is always greater or equal to 2 for x is a real number.

➡️Hence, Range = [2, ∞)

✴(iii) Given,✴

➡️f(x) = x, x is a real number

➡️Clearly, the range of f is the set of all real numbers.

➡️Thus,

➡️Range of f = R

Answer 2

Answer:-

(i)=>0<x

=>0<3x

=>−3x<0

=>2−3x<2

∴f(x)<2

∴range(−∞,2)

(ii)x²≥0

so x²+2≥2 ( adding on bs)

hence,

Range (2,∞)

(iii) f(x)=x,x is a real numbers

clearly , the range of f is the set of all real numbers

Range of f =R