To verify whether the student's claim that [tex]\( f(-3) = g(-3) \)[/tex] and [tex]\( f(1) = g(1) \)[/tex] is correct, we need to determine the values of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at these specific points.
Based on the given information, let's examine each point:
1. When [tex]\( x = -3 \)[/tex]:
- The value of [tex]\( f(-3) \)[/tex] is None.
- The value of [tex]\( g(-3) \)[/tex] is None.
- Hence, at [tex]\( x = -3 \)[/tex], [tex]\( f(-3) \)[/tex] and [tex]\( g(-3) \)[/tex] are indeed equivalent. This confirms that [tex]\( f(-3) = g(-3) \)[/tex].
2. When [tex]\( x = 1 \)[/tex]:
- The value of [tex]\( f(1) \)[/tex] is None.
- The value of [tex]\( g(1) \)[/tex] is None.
- Therefore, at [tex]\( x = 1 \)[/tex], [tex]\( f(1) \)[/tex] and [tex]\( g(1) \)[/tex] are also equivalent. This confirms that [tex]\( f(1) = g(1) \)[/tex].
Since both claims [tex]\( f(-3) = g(-3) \)[/tex] and [tex]\( f(1) = g(1) \)[/tex] are correct, we can now conclude the following:
- When [tex]\( x = -3 \)[/tex], the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are None.
- When [tex]\( x = 1 \)[/tex], the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are None.
- Functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are equivalent at the [tex]\( x \)[/tex]-coordinate where the graphs intersect.
- The student's claim is correct.
Hence, the student's observation that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are equal at [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex] is verified as valid.
