\# Which of the following exponential 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

To determine which of the given exponential functions is increasing and has a graph that is below the asymptote, we need to analyze each function. 1. **Identify the general form of the functions:** - Exponential functions can be expressed in the form $$ f(x) = a \cdot b^x + c $$, where: - $$ a $$ determines the vertical stretch or compression and the direction of the graph (positive or negative). - $$ b $$ is the base of the exponential function (if $$ b > 1 $$, the function is increasing; if $$ 0 < b < 1 $$, the function is decreasing). - $$ c $$ is the vertical shift (the horizontal asymptote is $$ y = c $$). 2. **Analyze each function:** - **a.** $$ f(x) = -\frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$ - Here, $$ a = -\frac{1}{2} $$ (negative), $$ b = \frac{1}{3} $$ (decreasing), and the asymptote is $$ y = -8 $$. - **b.** $$ f(x) = -\frac{1}{2}(3)^{x}-8 $$ - Here, $$ a = -\frac{1}{2} $$ (negative), $$ b = 3 $$ (increasing), and the asymptote is $$ y = -8 $$. - **c.** $$ f(x) = \frac{1}{2}\left(\frac{1}{3}\right)^{x}-8 $$ - Here, $$ a = \frac{1}{2} $$ (positive), $$ b = \frac{1}{3} $$ (decreasing), and the asymptote is $$ y = -8 $$. - **d.** $$ f(x) = \frac{1}{2}(3)^{x}-8 $$ - Here, $$ a = \frac{1}{2} $$ (positive), $$ b = 3 $$ (increasing), and the asymptote is $$ y = -8 $$. 3. **Determine the increasing functions and their positions relative to the asymptote:** - Functions **b** and **d** are increasing because they have a base $$ b > 1 $$. - Both functions have an asymptote at $$ y = -8 $$. Since they are increasing, they will eventually rise above the asymptote. 4. **Conclusion:** - The only function that is increasing and remains below the asymptote is **none** of the options, as both increasing functions (b and d) will eventually cross the asymptote. Thus, there is no function among the options that is increasing and remains below the asymptote. None of the functions are increasing and remain below the asymptote.

# Which of the following exponential
functions is increasing and with a graph that is
below the asymptote?
a.
b.
c.
d.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Pre Calculus Questions

# Which of the following exponential functions is increasing and with a graph that is below the asymptote? a. b. c. d.