Question

How do I solve this problem?The options for blank one are:- Y-intercept - X-interceptFor blank two:- decrease- increase- remain the same For blank three: - decrease- increase - remain the sameFor blank four: - 14For blank five:- 14

Answer 1

Consider the general function

`g(x)=a((1)/(2))^x`

Notice that, regardless of the value of a

`\lim _(x\to\infty)g(x)=a\lim _(x\to\infty)((1)/(2))^x=a\lim _(x\to\infty)(1)/(2^x)=a\cdot0=0`

Therefore, the graph cannot intersect the x-axis. Thus, the answer to the first blank is y-intercept.

Set b>4 and 1
`\begin{gathered} b((1)/(2))^0=b\cdot1=b\to(0,b)\text{ y-intercept} \\ and \\ c((1)/(2))^0=c\cdot1=c\to(0,c)\text{ y-intercept} \end{gathered}`Hence, the answer to the second blank is increase, and the answer to the third blank is decrease.

The descent of the graph depends on the derivative of the function; therefore,

`g^(\prime)(x)=a(-2^(-x)\log 2)=-a\log 2(2^(-x))`

Notice that there is a 'a' term in the derivative; thus, the greater a is, the steeper is the descent of the graph.

Then, the answer to the fourth blank is a>4, and the answer to the fifth blank is 1