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

enter image description here enter image description here enter image description here

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 $f\left(\frac{1}{5}x\right)$.

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.