The graph of the absolute value parent function, [tex]f(x)=|x|[/tex], is stretched horizontally by a factor of 5 to create the graph of [tex]g(x)[/tex]. What function is [tex]g(x)[/tex]?
A. [tex]g(x)=5|x|[/tex] B. [tex]g(x)=|x+5|[/tex] C. [tex]g(x)=\left|\frac{1}{5} x\right|[/tex] D. [tex]g^{\prime}(x)=|5 x|[/tex]
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]
