To apply a horizontal shrink by a factor of $\frac{1}{3}$ to the function $f(x) = |x - 1| + 3$, we need to adjust the input variable in such a way that the function is "compressed" horizontally.

Horizontal Shrink:

• A horizontal shrink by a factor of $k$ (where $k > 1$) replaces $x$ with $\frac{x}{k}$.
• In this case, the shrink factor is $\frac{1}{3}$, so we replace $x$ with $\frac{x}{\frac{1}{3}} = 3x$.

Step-by-step transformation:

The original function is $f(x) = |x - 1| + 3$.

• To apply the horizontal shrink, replace $x$ with $3x$ in the expression: $f(3x) = |3x - 1| + 3$

So, the new function resulting from applying a horizontal shrink by a factor of $\frac{1}{3}$ to $f(x) = |x - 1| + 3$ is: $g(x) = |3x - 1| + 3$

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Would you like a further explanation or example? Let me know if you'd like any steps clarified!

Here are some related questions for you to explore further:

1. What happens if we apply a horizontal stretch instead of a shrink to $f(x) = |x - 1| + 3$?
2. How does a vertical shift affect the graph of a function like $f(x) = |x - 1| + 3$?
3. How do horizontal transformations like shrinks and stretches relate to the standard form of absolute value functions?
4. What is the effect of applying a horizontal shrink by a factor of 2 on the function $f(x) = |x - 5|$?
5. How can you combine horizontal and vertical transformations on functions like absolute value?

Tip: When applying horizontal transformations, remember that the input variable $x$ is affected by the transformation, not the output of the function.