Let *f* and *g* be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function:

(vi) \[2f - \sqrt{5} g\]

Advertisement Remove all ads

#### Solution

Given:

\[f\left( x \right) = \sqrt{x + 1}\text{ and } g\left( x \right) = \sqrt{9 - x^2}\]

Clearly,

\[f\left( x \right) = \sqrt{x + 1}\] is defined for all

Thus, domain (

Again,

*x*≥ - 1.Thus, domain (

*f*) = [1, ∞]Again,

\[g\left( x \right) = \sqrt{9 - x^2}\] is defined for 9 -

*x*^{2}≥ 0 ⇒*x*^{2}- 9 ≤ 0⇒

*x*^{2}- 3^{2}≤ 0⇒ (

*x*+ 3)(*x - 3) ≤ 0*

*⇒*\[x \in \left[ - 3, 3 \right]\]Thus, domain (

*g*) = [ - 3, 3]Now,

domain (

*f*) ∩ domain(*g*) = [ -1, ∞] ∩ [- 3, 3] = [ -1, 3](vi) \[2f - \sqrt{5}g: \left[ - 1, 3 \right] \to \text{ R is given by } \left( 2f - \sqrt{5}g \right)\left( x \right) = 2\sqrt{x + 1} - \sqrt{5}\left( \sqrt{9 - x^2} \right)\] \[= 2\sqrt{x + 1} - \sqrt{45 - 5 x^2}\]

Concept: Concept of Functions

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads