Question

homework assignment 6.2 graphs of exponential f
due sunday by 11:59pm points 10 submitting an external tool
homework assignment 6.2 graphs of exponential functions
score: 4.4/10 answered: 5/10
question 6
describe the long run behavior of ( f(x)=4(4)^{x}+2 ):
as ( x
ightarrow-infty, f(x)
ightarrow ) ?
as ( x
ightarrow infty, f(x)
ightarrow ) ?
question help: video read written example

Explanation:

Step1: Analyze the behavior as \(x\to-\infty\)

For the exponential function \(y = a\cdot b^{x}\), when \(b> 1\) (here \(b = 4\)) and \(x\to-\infty\), \(b^{x}=\frac{1}{b^{-x}}\to0\).
So for \(f(x)=4(4)^{x}+2\), as \(x\to-\infty\), \(4(4)^{x}\to0\). Then \(f(x)=4(4)^{x}+2\to0 + 2\).

Step2: Analyze the behavior as \(x\to\infty\)

When \(b>1\) (here \(b = 4\)) and \(x\to\infty\), \(b^{x}\to\infty\).
For \(f(x)=4(4)^{x}+2\), as \(x\to\infty\), \(4(4)^{x}\to\infty\). Then \(f(x)=4(4)^{x}+2\to\infty+ 2\).

Answer:

As \(x\to-\infty\), \(f(x)\to2\); as \(x\to\infty\), \(f(x)\to\infty\)