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]
