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Home » Let f(x) = 2x + 5 and g(x) = x2 + x. Describe (iii) fg (iv) f/g Find the domain in each case.
Let f(x) = 2x + 5 and g(x) = x2 + x. Describe (iii) fg (iv) f/g Find the domain in each case.
CBSE | Class 11 | Exercise 3.4 | Functions | Maths | RD Sharma
Let f(x) = 2x + 5 and g(x) = x2 + x. Describe (iii) fg (iv) f/g Find the domain in each case.

Solution:

(iii) fg

We know that: (fg)(x) = f(x)g(x)

\left( fg \right)\left( x \right)=\text{ }\left( 2x\text{ }+\text{ }5 \right)\left( {{x}^{2}}~+\text{ }x \right)

\left( fg \right)\left( x \right)=\text{ }2x\left( {{x}^{2}}~+\text{ }x \right)\text{ }+\text{ }5\left( {{x}^{2}}~+\text{ }x \right)

\left( fg \right)\left( x \right)=\text{ }2{{x}^{3}}~+\text{ }2{{x}^{2}}~+\text{ }5{{x}^{2}}~+\text{ }5x

\left( fg \right)\left( x \right)=\text{ }2{{x}^{3}}~+\text{ }7{{x}^{2}}~+\text{ }5x

(fg)(x) is defined for x belonging to R. Therefore, the domain of fg is R

(iv) f/g

We know that:

\left( f/g \right)\text{ }\left( x \right)=\text{ }f\left( x \right)/g\left( x \right)\text{ }

\left( f/g \right)\text{ }\left( x \right)=\text{ }\left( 2x+5 \right)/\left( {{x}^{2}}+x \right)
(f/g) (x) is defined for all real values of x, except for the case when x2 + x = 0.

{{x}^{2}}~+\text{ }x\text{ }=\text{ }0x\left( x\text{ }+\text{ }1 \right)\text{ }

{{x}^{2}}~+\text{ }x\text{ }=0\text{ }or\text{ }x\text{ }+\text{ }1\text{ }=\text{ }0

x = 0 or –1

When x = 0 or –1, The outcome of the division will be ambiguous, hence (f/g) (x)will be undefined.

∴ The domain of f/g = R – {–1, 0}

i

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