Question

select the correct answer.
exponential function ( f ) is represented by the table.

( x )	-1	0	1	2	3
( f(x) )	78	24	6	0	-2

function ( g ) is an exponential function passing through the points ( (0,15) ) and ( (2,0) ).
which statement correctly compares the behavior of the two functions on the interval ( (0, 2) )?
a. both functions are positive on the interval, but one function is increasing while the other is decreasing
b. both functions are positive and increasing on the interval.
c. both functions are positive and decreasing on the interval.
d. one function is positive on the interval, while the other is negative.

Explanation:

Step1: Analyze function $f$ on $(0,2)$

From the table, at $x=1$, $f(1)=6>0$; at $x=2$, $f(2)=0$. On $(0,2)$, $f(x)$ goes from 24 to 6 to 0, so it is positive and decreasing.

Step2: Find equation of function $g$

General exponential form: $g(x)=ab^x$. Use $(0,15)$: $15=ab^0 \implies a=15$. Use $(2,0)$: $0=15b^2$, which is not a valid exponential function, but we check values on $(0,2)$: at $x=0$, $g(0)=15>0$; at $x=2$, $g(2)=0$. For exponential decay (since it goes from 15 to 0), $g(x)$ is positive and decreasing on $(0,2)$.

Step3: Compare behaviors

Both functions are positive and decreasing on $(0,2)$.

Answer:

C. Both functions are positive and decreasing on the interval.