16. The function [tex]$f(x)=|x|$[/tex] is multiplied by [tex]$k$[/tex] to create the new function [tex]$g(x)=k|x|$[/tex]. Which statement is true about the graphs of [tex][tex]$f(x)$[/tex][/tex] and [tex]$g(x)$[/tex] if [tex]$k=\frac{1}{2}$[/tex]?

1. [tex][tex]$g(x)$[/tex][/tex] is a reflection of [tex]$f(x)$[/tex] over the [tex][tex]$y$[/tex]-axis[/tex].
2. [tex]$g(x)$[/tex] is a reflection of [tex]$f(x)$[/tex] over the [tex][tex]$x$[/tex]-axis[/tex].
3. [tex]$g(x)$[/tex] is wider than [tex]$f(x)$[/tex].
4. [tex][tex]$g(x)$[/tex][/tex] is narrower than [tex]$f(x)$[/tex].



Answer :

To determine how the function [tex]\( g(x) \)[/tex] relates to the function [tex]\( f(x) \)[/tex] when [tex]\( k = \frac{1}{2} \)[/tex], let's analyze the transformation step-by-step.

1. Original Function: The given function is [tex]\( f(x) = |x| \)[/tex]. This is the absolute value function, which forms a "V" shape with its vertex at the origin (0,0) and opens upwards.

2. New Function: The new function is [tex]\( g(x) = k|x| \)[/tex], where [tex]\( k = \frac{1}{2} \)[/tex]. Substituting the value of [tex]\( k \)[/tex], we get:
[tex]\[ g(x) = \frac{1}{2} |x| \][/tex]

3. Transformation Analysis: Since [tex]\( k = \frac{1}{2} \)[/tex] is a positive value less than 1, multiplying [tex]\( |x| \)[/tex] by [tex]\(\frac{1}{2}\)[/tex] scales the function vertically, reducing each [tex]\( y \)[/tex]-value to half of its original value.

4. Graph Behavior: To understand how the graph changes, consider a few points on [tex]\( f(x) \)[/tex] and how they transform under [tex]\( g(x) \)[/tex]:
- At [tex]\( x = 1 \)[/tex], [tex]\( f(x) = |1| = 1 \)[/tex] and [tex]\( g(x) = \frac{1}{2}|1| = \frac{1}{2} \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( f(x) = |2| = 2 \)[/tex] and [tex]\( g(x) = \frac{1}{2}|2| = 1 \)[/tex]

Observing other points similarly, we see that [tex]\( g(x) \)[/tex] has [tex]\( y \)[/tex]-values that are half the corresponding [tex]\( y \)[/tex]-values of [tex]\( f(x) \)[/tex].

5. Conclusion: Since each point on [tex]\( g(x) \)[/tex] has a lower [tex]\( y \)[/tex]-value compared to [tex]\( f(x) \)[/tex] (but the [tex]\( x \)[/tex]-values remain the same), the effect is that the graph of [tex]\( g(x) \)[/tex] is "wider" than the graph of [tex]\( f(x) \)[/tex].

Thus, the transformation makes the graph of [tex]\( g(x) \)[/tex] wider than the graph of [tex]\( f(x) \)[/tex], because the steepness of the "V" shape decreases when all [tex]\( y \)[/tex]-values are reduced.

6. Correct Statement:
The correct statement about the graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
(3) [tex]\( g(x) \)[/tex] is wider than [tex]\( f(x) \)[/tex].

Therefore, the answer is:
[tex]\[ \boxed{3} \][/tex]