First slide

Functions (XII)

Question

The domain of the function $f (x) = \log_{10} [1 - \log_{10} (x^{2} - 5 x + 16)] is$

Moderate

A

(2, 3)
B

[2, 3]
C

(2, 3]
D

[2, 3)

Solution

f (x) is defined if
$\begin{array}{l} 1 - \log_{10} (x^{2} - 5 x + 16) > 0 and x^{2} - 5 x + 16 > 0 \\ \Rightarrow \log_{10} (x^{2} - 5 x + 16) < 1 and {(x - \frac{5}{2})}^{2} + \frac{39}{4} > 0 \\ \Rightarrow x^{2} - 5 x + 16 < 10^{1} = 10 [∵ {(x - \frac{5}{2})}^{2} + \frac{39}{4} > 0 for all real x] \end{array}$
⇒ x2 – 5x + 6 < 0
⇒ (x – 3) (x – 2) < 0 ⇒ 2 < x < 3
$∴$ Domain of f = (2, 3)

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