Question

q 3
which of the following statements are true about the graph of $f(x)=0.4(5)^{x-1}+2$?
i. the functions inverse is decreasing.
ii. as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to 2$.
iii. the y-intercept is $(0, 2)$.
iv. the asymptote is $y = 2$.
a i only
b ii and iv only
c i and iii only
d ii, iii, and iv only

Explanation:

Step1: Analyze the original function

The function is $f(x)=0.4(5)^{x-1}+2$. The base of the exponential term is $5>1$, so $f(x)$ is an increasing exponential function. The inverse of an increasing function is increasing, so Statement I is false.

Step2: Evaluate end behavior

As $x\to\infty$: $5^{x-1}\to\infty$, so $f(x)=0.4(5)^{x-1}+2\to\infty$.
As $x\to-\infty$: $5^{x-1}=\frac{1}{5^{1-x}}\to0$, so $f(x)=0.4(0)+2\to2$.
Statement II is true.

Step3: Calculate y-intercept

Set $x=0$:

$$\begin{align*} f(0)&=0.4(5)^{0-1}+2\\ &=0.4\times\frac{1}{5}+2\\ &=0.08+2\\ &=2.08 \end{align*}$$

The y-intercept is $(0, 2.08)$, so Statement III is false.

Step4: Identify horizontal asymptote

For exponential functions of the form $a(b^{x-h})+k$, the horizontal asymptote is $y=k$. Here $k=2$, so the asymptote is $y=2$. Statement IV is true.

Answer:

B. II and IV only