Question

identify the end behavior of the given functions. select the correct answer from each drop - down menu. the function ( m(x)=2^{x}+1 ) increases toward the line ( y = 1 ) as ( x ) increases. the function ( n(x)=10^{x - 3}+5 ) decreases toward the line ( y = 5 ) as ( x ) decreases. the function ( p(x)=-4^{x}-2 ) decreases toward as ( x ) decreases.

Explanation:

Step1: Analyze \( m(x)=2^{x}+1 \)

For an exponential function \( y = a^{x}\) (\(a>1\), here \(a = 2\)), as \(x\) increases, \(2^{x}\) increases. So \(m(x)=2^{x}+1\) increases toward the line \(y = 1\) (horizontal - asymptote concept: \(\lim_{x
ightarrow-\infty}2^{x}+1=1\)) as \(x\) decreases and increases as \(x\) increases.

Step2: Analyze \( n(x)=10^{x - 3}+5 \)

For the exponential function \(y = 10^{x-3}+5\) (\(a = 10>1\)), as \(x\) decreases, \(10^{x-3}\) approaches \(0\). So \(n(x)=10^{x - 3}+5\) decreases toward the line \(y = 5\) (horizontal - asymptote: \(\lim_{x
ightarrow-\infty}10^{x-3}+5 = 5\)) as \(x\) decreases.

Step3: Analyze \( p(x)=-4^{x}-2 \)

For the exponential function \(y=-4^{x}-2\) (\(a = 4>1\)), as \(x\) decreases, \(4^{x}\) approaches \(0\). So \(p(x)=-4^{x}-2\) increases toward \(- 2\) (horizontal - asymptote: \(\lim_{x
ightarrow-\infty}-4^{x}-2=-2\)) as \(x\) decreases.

Answer:

For \(m(x)=2^{x}+1\): increases toward the line \(y = 1\) as \(x\) decreases and increases as \(x\) increases.
For \(n(x)=10^{x - 3}+5\): decreases toward the line \(y = 5\) as \(x\) decreases.
For \(p(x)=-4^{x}-2\): increases toward \(-2\) as \(x\) decreases.