Question

f(x) = \frac{1}{x}
the graph of the function has a vertical asymptote of x =
the graph of the function has a horizontal asymptote of y =

Explanation:

Step1: Find Vertical Asymptote

For the function \( f(x)=\frac{1}{x} \), vertical asymptotes occur where the denominator is zero (and numerator non - zero). Here, denominator \( x = 0 \) makes the function undefined (numerator \( 1
eq0 \)), so vertical asymptote is at \( x = 0 \).

Step2: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{a}{x^n}+b \) (in this case \( a = 1 \), \( n = 1 \), \( b = 0 \)), as \( x\to\pm\infty \), \( \frac{1}{x}\to0 \). So the horizontal asymptote is \( y = 0 \).

Answer:

The graph of the function has a vertical asymptote of \( x=\boldsymbol{0} \)
The graph of the function has a horizontal asymptote of \( y=\boldsymbol{0} \)