The domain of definition of the functions y(x) given by the equation a x +ay=a(a1) is

Question

The domain of definition of the functions y(x) given by the equation a x +ay=a(a>1) is (A) 0< x ≤ 1 (B) 0 ≤ x ≤ 1 (C) -∞< x< 1 (D) -∞< x ≤ 0

Tardigrade Admin · Accepted Answer

ax-1+ay-1=1 ⇒ ay-1=1-ax-1 ⇒(y-1) log a= log (1-ax-1) text R.H.S is defined if 1-ax-1>0 ⇒ ax-1<1=a° ⇒ x-1< 0 text as a >1 ⇒ x< 1 text and the required domain is -∞ < x< 1 text .

Q. The domain of definition of the functions $y (x)$ given by the equation $a^{x} + a^{y} = a (a > 1)$ is

$0 < x \leq 1$

$0 \leq x \leq 1$

$- \infty < x < 1$

$- \infty < x \leq 0$

$a^{x - 1} + a^{y - 1} = 1$
$\Rightarrow a^{y - 1} = 1 - a^{x - 1} \Rightarrow (y - 1) lo g a = lo g (1 - a^{x - 1})$
$R.H.S is defined if 1 - a^{x - 1} > 0$
$\Rightarrow a^{x - 1} < 1 = a^{\circ} \Rightarrow x - 1 < 0 as a > 1$
$\Rightarrow x < 1 and the required domain is - \infty < x < 1 .$

Q. The domain of definition of the functions y(x) given by the equation ax+ay=a(a>1) is

0<x≤1

0≤x≤1

−∞<x<1

−∞<x≤0