Q. 24.5( 2 Votes )

# Let f(x) = 2x + 5 and g(x) = x2 + x. Describei. f + gii. f – giii. fgiv. Find the domain in each case.

Given f(x) = 2x + 5 and g(x) = x2 + x

Clearly, both f(x) and g(x) are defined for all x R.

Hence, domain of f = domain of g = R

i. f + g

We know (f + g)(x) = f(x) + g(x)

(f + g)(x) = 2x + 5 + x2 + x

(f + g)(x) = x2 + 3x + 5

Clearly, (f + g)(x) is defined for all real numbers x.

The domain of (f + g) is R

ii. f – g

We know (f – g)(x) = f(x) – g(x)

(f – g)(x) = 2x + 5 – (x2 + x)

(f – g)(x) = 2x + 5 – x2 – x

(f – g)(x) = 5 + x – x2

Clearly, (f – g)(x) is defined for all real numbers x.

The domain of (f – g) is R

iii. fg

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

(fg)(x) = (2x + 5)(x2 + x)

(fg)(x) = 2x(x2 + x) + 5(x2 + x)

(fg)(x) = 2x3 + 2x2 + 5x2 + 5x

(fg)(x) = 2x3 + 7x2 + 5x

Clearly, (fg)(x) is defined for all real numbers x.

The domain of fg is R

iv.

We know

Clearly, is defined for all real values of x, except for the case when x2 + x = 0.

x2 + x = 0

x(x + 1) = 0

x = 0 or x + 1 = 0

x = 0 or –1

When x = 0 or –1, will be undefined as the division result will be indeterminate.

Thus, domain of = R – {–1, 0}

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