First slide

Functions (XII)

Question

Let f be a function satisfying $f (x) = \frac{a^{x}}{a^{x} + \sqrt{a}} = g_{a} (x) (a > 0)$
On the basis of above information, answer the following questions

Moderate

Question

Let $f (x) = g_{9} (x)$ , then the value of $[\sum_{r = 1}^{1995} f (\frac{r}{1996})]$ is (where [.] denotes the greatest integer function)

A

995
B

996
C

997
D

998

Solution

$f (x) = g_{9} (x) =$ $\frac{9^{x}}{9^{x} + \sqrt{9}} = \frac{9^{x}}{9^{x} + 3}$
$t h e r e f o r e f (x) + f (1 - x) = 1$
$\sum_{r = 1}^{1995} f (\frac{r}{1996}) = f (\frac{1}{1996}) + f (\frac{2}{1996}) + f (\frac{3}{1996}) + \dots + f (\frac{1993}{1996}) + f (\frac{1994}{1996}) + f (\frac{1995}{1996})$
$= \{f (\frac{1}{1996}) + f (\frac{1995}{1996})\} + \{f (\frac{2}{1996}) + f (\frac{1994}{1996})\} + \{f (\frac{3}{1996}) + f (\frac{1993}{1996})\} + \dots + \{f (\frac{997}{1996}) + f (\frac{999}{1996})\} + f (\frac{998}{1996})$
$\begin{aligned} = \underset{997 times}{\underset{⏟}{1 + 1 + 1 \dots + 1 + 1}} + f (\frac{1}{2}) \\ = 997 + \frac{1}{2} \\ = 997 .5 \end{aligned}$

Question

Let f(x) = g₄(x), the $\sum_{r = 1}^{1996} f (\frac{r}{1997})$ is

A

zero
B

even
C

odd
D

None of these

Solution

$\sum_{r = 1}^{1996} f (\frac{r}{1997}) = f (\frac{1}{1997}) + f (\frac{2}{1997}) + f (\frac{3}{1997}) + \dots + f (\frac{1994}{1997}) + f (\frac{1995}{1997}) + f (\frac{1996}{1997})$
$= \{f (\frac{1}{1997}) + f (\frac{1996}{1997})\} + \{f (\frac{2}{1997}) + f (\frac{1995}{1997})\} + \{f (\frac{3}{1996}) + f (\frac{1994}{1996})\} + \dots$
=1+1+1+... 998 times or = 998 (even)

Question

The value of $g_{5} (x) + g_{5} (1 - x)$ is

A

1
B

5
C

10
D

None of these

Solution

$g_{5} (x) + g_{5} (1 - x) = f (x) + f (1 - x) = 1$

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