Question

the zeros of a function $f(x)$ are 3 and 4, while the zeros of a second function $g(x)$ are 3 and 5. what are the zero(s) of the function $y = \frac{f(x)}{g(x)}$?
\bigcirc a) $x = 4$
\bigcirc b) $x = 4, x = 5$
\bigcirc c) $x = 3$
\bigcirc d) $x = 3, x = 4, x = 5$

Explanation:

Step1: Recall Zero of Rational Function

The zero of a rational function \( y = \frac{f(x)}{g(x)} \) occurs where \( f(x) = 0 \) and \( g(x)
eq 0 \).

Step2: Find Zeros of \( f(x) \)

Given zeros of \( f(x) \) are \( 3 \) and \( 4 \), so \( f(3) = 0 \) and \( f(4) = 0 \).

Step3: Check \( g(x) \) at These Zeros

Zeros of \( g(x) \) are \( 3 \) and \( 5 \), so \( g(3) = 0 \) and \( g(5)
eq 0 \), \( g(4)
eq 0 \) (since \( 4 \) is not a zero of \( g(x) \)).

Step4: Determine Valid Zeros

For \( x = 3 \), \( g(3) = 0 \), so \( \frac{f(3)}{g(3)} \) is undefined. For \( x = 4 \), \( f(4) = 0 \) and \( g(4)
eq 0 \), so \( x = 4 \) is a zero.

Answer:

a) \( x = 4 \)