To find the function [tex]\( g(x) \)[/tex] given that the absolute value parent function [tex]\( f(x) = |x| \)[/tex] is stretched horizontally by a factor of 5, follow these steps:
1. Understanding Horizontal Stretch:
A horizontal stretch by a factor of 5 means that every [tex]\( x \)[/tex]-coordinate on the graph of [tex]\( f(x) \)[/tex] is stretched out by a factor of 5. In other words, each [tex]\( x \)[/tex]-value on the graph of [tex]\( f \)[/tex] is replaced by [tex]\( x/5 \)[/tex] on the graph of [tex]\( g \)[/tex].
2. Applying the Stretch to the Function:
To stretch [tex]\( f(x) = |x| \)[/tex] horizontally by a factor of 5, we replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( x/5 \)[/tex]. This gives us:
[tex]\[
g(x) = f\left(\frac{x}{5}\right)
\][/tex]
Since [tex]\( f(x) = |x| \)[/tex], we substitute [tex]\( x/5 \)[/tex] into the absolute value function:
[tex]\[
g(x) = \left| \frac{x}{5} \right|
\][/tex]
3. Simplifying the Expression:
The expression [tex]\( \left| \frac{x}{5} \right| \)[/tex] can be simplified slightly:
[tex]\[
g(x) = \frac{1}{5} |x|
\][/tex]
4. Validating with Options:
Now we match our result with the provided options. The function [tex]\( g(x) = \frac{1}{5} |x| \)[/tex] corresponds to option C.
Therefore, the function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = \left| \frac{1}{5} x \right|
\][/tex]
Hence, the correct choice is:
C. [tex]\( g(x) = \left| \frac{1}{5} x \right| \)[/tex]
