Question

what happens to the graph of an the exponential function ( f(x)=2^{x} ) as ( x ) approaches positive infinity and negative infinity? as ( x ) approaches positive infinity, the graph fluctuates in value, and as ( x ) approaches negative infinity, the graph becomes undefined. as ( x ) approaches positive infinity, the graph approaches the ( y )-axis, and as ( x ) approaches negative infinity, the graph moves away from the ( y - )axis. as ( x ) approaches positive infinity, the graph stabilizes, and as ( x ) approaches negative infinity, it dips into the negatives. as ( x ) approaches positive infinity, the graph increases without bound, and as ( x ) approaches negative infinity, the graph approaches zero.

Explanation:

For the exponential function \( f(x)=2^{x} \), when \( x\to+\infty \), since the base \( a = 2>1 \), \( y = 2^{x}\) grows without bound. When \( x\to-\infty\), we can rewrite \( y=2^{x}=\frac{1}{2^{-x}}\). As \( x\to-\infty\), \( -x\to+\infty\), and \( 2^{-x}\to+\infty\), so \( y = 2^{x}=\frac{1}{2^{-x}}\to0\).

Answer:

As \( x \) approaches positive infinity, the graph increases without bound, and as \( x \) approaches negative infinity, the graph approaches zero.