Write the domain of the following real functions?(a) \(\sqrt{9-x^2}\)(

1.	Write the domain of the following real functions?(a) \(\sqrt{9-x^2}\)(b) 10x (c) \(\frac{2}{4x+7}\)(d) log (2 – 3x)
Answer» (i) Since f is a real function, 9 – x² ≥ 0 ⇒ (3 + x) (3 – x) ≥ 0 ⇒ – a ≤ x ≤ a ∴ Domain of f (x) = \(\sqrt{9-x^2}\) is {x\| – 3 ≤ x ≤ 3}. (ii) Let f (x) = 10^x a^x is defined for all real values of x when a > 0 Here a = 10, ∴ Domain of f (x) = 10^x is R. (iii) f (x) = \(\frac{2}{4x+7}\) Since f is real, x can take all real values except the value for which 4x + 7 = 0, i.e., x = \(-\frac{7}{4}\) ⇒x ≠ \(-\frac{7}{4}\) ∴ Domain of f (x) = \(\frac{2}{4x+7}\) = R - { \(-\frac{7}{4}\) }. (iv) f (x) = log (2 – 3x) For f (x) to be defined (2 – 3x) > 0 ⇒ 2 > 3x ⇒ x < \(\frac{2}{3}\) ∴ Domain of f (x) = log (2 – 3x) = −∞< x < \(\frac{2}{3}\) or x \(\in\) (−∞ \(\frac{2}{3}\) )

Write the domain of the following real functions?(a) \(\sqrt{9-x^2}\)(b) 10x (c) \(\frac{2}{4x+7}\)(d) log (2 – 3x)

Answer»

(i) Since f is a real function,

9 – x² ≥ 0

⇒ (3 + x) (3 – x) ≥ 0

⇒ – a ≤ x ≤ a

∴ Domain of f (x) = \(\sqrt{9-x^2}\) is {x| – 3 ≤ x ≤ 3}.

(ii) Let f (x) = 10^x

a^x is defined for all real values of x when a > 0 Here a = 10,

∴ Domain of f (x) = 10^x is R.

(iii) f (x) = \(\frac{2}{4x+7}\)

Since f is real, x can take all real values except the value for which 4x + 7 = 0,

i.e., x = \(-\frac{7}{4}\)

⇒x ≠ \(-\frac{7}{4}\)

∴ Domain of f (x) = \(\frac{2}{4x+7}\) = R - { \(-\frac{7}{4}\) }.

(iv) f (x) = log (2 – 3x)

For f (x) to be defined (2 – 3x) > 0

⇒ 2 > 3x

⇒ x < \(\frac{2}{3}\)

∴ Domain of f (x) = log (2 – 3x) = −∞< x < \(\frac{2}{3}\)

or x \(\in\) (−∞ \(\frac{2}{3}\) )