# Function transformation: shrink horizontally

Write the formula for $$f(x)$$, if the graph of $$f$$ can be obtained from the graph of $$y = g(x)$$ by shrink horizontally by a factor of $$5$$ then shift left $$3$$ units The equation should be $$f(x) = g(5(x+3))$$ or $$g\left(\frac{1}{5}(x+3)\right)$$? I prefer the second answer but my teacher said the correct is the first one? Can anyone explain for me why it is $$5$$ instead of $$\frac{1}{5}$$ while we are dealing with horizontal shrinking? Thanks a lot

I found this counterintuitive when I was first learning algebra too.

Think about it like this: $f(5x)$ gives you $f(0)$ at $x=0$, then $f(5)$ at $x=1$, then $f(10)$ at $x=2$. Varying the input parameter from $0$ to $2$ made the function go all the way from $f(0)$ to $f(10)$. So the section of the graph of $f(x)$ that used to have width 10 will have only width 2 in the graph of $f(5x)$.

If that still doesn't click, I would just suggest drawing out a bunch of explicit examples for different functions $f$.

Intuitively, a function that's shrunk covers its original range values on a shorter interval. With $$5x$$ instead of $$x$$, consider the original function on the interval $$[0,1]$$, you get $$5\cdot(1/5) =1$$, so that the function covered all its original values on $$[0,1]$$ by the time you get to $$x=1/5$$, i.e on the interval $$[0,1/5]$$. In general then it covers its range 5 times faster.

Another easy way is to consider $$cx$$ for $$c$$ getting really large. Then for small $$x$$, you've already covered a huge portion of the function's range.

To shrink a function means to make the graph of the function seems narrower.

For example, consider the function $$f(x)=x^2$$ If you want to make the function shrink horizontally by a factor of 2 you would want the function $$f(2x) = (2x)^2 = 4x^2$$ On the other hand, you would argue that $$f\left(\frac{1}{2}x\right) = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$$ is correct.

If you graph the functions, you would get

Obviously, the function $$f(2x) = (2x)^2 = 4x^2$$ seems narrower. Similarly, your question is asking you to shrink the function by a factor of five, so it should be $$f(5x)$$ instead of $$f\left(\frac{1}{5}x\right)$$.