Answer: $$ f(x)=-\frac{1}{2}(3)^{x}-8 $$ functions is increasing and with a graph that is below the asymptote? a. $$ f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$ b. $$ f(x)=-\frac{1}{2}(3)^{x}-8 $$ c. $$ f(x)=\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$ d. $$ f(x)=\frac{1}{2}(3)^{x}-8 $$

Question

UpStudy AI · Accepted Answer

We wish to select the function $$ f(x)=a\cdot b^x + k $$ that satisfies both: 1. The function is increasing. 2. Its graph stays entirely below the horizontal asymptote $$ y=k $$. For an exponential function $$ f(x)=a\cdot b^x+k $$: • The horizontal asymptote is $$ y=k $$. • The sign of the coefficient $$ a $$ and the value of $$ b $$ determine the behavior: – If $$ b>1 $$ and $$ a>0 $$, the function is increasing and its outputs are above the asymptote (if it approaches $$ k $$ from below, then eventually the function must cross it or lie above it). – If $$ 00 $$, the function is decreasing (approaching the asymptote from above for $$ k $$). – If $$ 00 $$ for all $$ x $$, we have $$ a\cdot b^x<0, $$ so adding $$ k $$ gives $$ f(x)=a\cdot b^x+k1 $$ and $$ a=-\frac{1}{2} $$. Its derivative is \[ f'(x)=-\frac{1}{2}(3)^{x}\ln(3). $$ Since $$ \ln(3)>0 $$, $$ f'(x)<0 $$; the function is decreasing (even though $$ f(x)<-8 $$ for all $$ x $$). It does not satisfy the increasing condition. **Option c:** $$ f(x)=\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8. $$ Here $$ a=\frac{1}{2}>0 $$ and $$ b=\frac{1}{3} $$ (with $$ 0<\frac{1}{3}<1 $$). Its derivative is $$ f'(x)=\frac{1}{2}\left(\frac{1}{3}\right)^{x}\ln\left(\frac{1}{3}\right). $$ Since $$\ln\left(\frac{1}{3}\right)<0$$, we have $$ f'(x)<0 $$; the function is decreasing. Moreover, the exponential term is positive so for some $$ x $$ the function can be above the asymptote $$ -8 $$. **Option d:** $$ f(x)=\frac{1}{2}(3)^{x}-8. $$ Here $$ a>0 $$ and $$ b>1 $$ so the function is increasing. However, since $$ \frac{1}{2}(3)^{x}>0 $$ for all $$ x $$, we have $$ f(x)>-8 \quad \text{for all } x, $$ so its graph lies above the asymptote $$ y=-8 $$. Only option a satisfies both conditions of being increasing and having the graph entirely below the horizontal asymptote. Thus, the correct answer is a. $$ f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$. The correct answer is option a: $$ f(x)=-\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$.

Answer:
functions is increasing and with a graph that is
below the asymptote?
a.
b.
c.
d.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Calculus Questions

Answer: functions is increasing and with a graph that is below the asymptote? a. b. c. d.