A student graphs the functions [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex]. The student claims that [tex]$f(-3)=g(-3)$[/tex] and [tex][tex]$f(1)=g(1)$[/tex][/tex]. Is the student's claim correct? Use the drop-down menus to complete the statements.

Click the arrows to choose an answer from each menu.

When [tex]$x=-3$[/tex], the values of [tex]$f(x)$[/tex] and [tex][tex]$g(x)$[/tex][/tex] are Choose...

When [tex]$x=1$[/tex], the values of [tex]$f(x)$[/tex] and [tex][tex]$g(x)$[/tex][/tex] are Choose...

Functions [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] are equivalent at the [tex][tex]$x$[/tex][/tex]-coordinate where the graphs intersect Choose... [tex]$\square$[/tex] .

The student's claim Choose... correct.



Answer :

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.