Question

identify the horizontal asymptote of each graph.
graph of f(x) = 6^x with options y=0, y=1, y=6

Explanation:

Step1: Recall exponential function properties

For an exponential function \( f(x) = a^x \) (where \( a>0, a
eq1 \)), the horizontal asymptote is determined by the limit as \( x \to -\infty \).
\( \lim_{x \to -\infty} a^x = 0 \) (since \( a^x=\frac{1}{a^{|x|}} \), and as \( |x| \to \infty \), \( a^{|x|} \to \infty \) for \( a > 1 \), so \( \frac{1}{a^{|x|}} \to 0 \)).

Step2: Apply to \( f(x)=6^x \)

Here, \( a = 6>1 \). As \( x \to -\infty \), \( 6^x=\frac{1}{6^{|x|}} \to 0 \). From the graph, we also see the curve approaches the x - axis (where \( y = 0 \)) as \( x \) becomes very negative.

Answer:

A. \( y = 0 \)