Horizontal shrink / stretch

Question

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(1/5(x+3))?

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 1/5 while we are dealing with horizontal shrinking? 

Thanks a lot

Answer 1

Let f(x) = √x , for example.

f(16) = √16 = 4

Let g(x) = f[(1/2)x], that is B = 1/2. We expect a horizontal stretch by a factor of 2. What is the value of x that generates a y-value of 4?

√ [(1/2)x] =? 4

(1/2)x = 16 (square both sides)

x = 32 , so x went from 16 on the parent to 32 on the transformed function to generate a y-value of 4.

Because we input (1/2)x into the f function it takes "longer" to get to the x-value that generates the y-value without that (1/2) multiplier.

Now consider h(x) = f(2x), i.e. B = 2. We expect a horizontal shrink by a factor of 1/2. What value of x will give a y-value of 4?

√(2x) = ? 4

2x = 16 (square both sides)

x = 8, so x went from 16 on the parent to x equal 8 on the transformed function to generate the same y-value (4). We got to the required input to f more quickly because the x input was doubled.

Experiment here:

desmos.com/calculator/90e561dce8