C1 Graph transformations of 1/x graphs

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You have the $f(x) = \frac{3}{x}$. Then $f(x+2) = \frac{3}{x+2}$ shifts the graph two units to the left, so there will be one intersection with the y-axis.

This is given when $f(0+2) = \frac{3}{0 + 2} = f(2) = \cdots$. The shift is horizontal and hence there is still no x-axis intersections.

so it doesn't matter where the x is in this one? so if this function has a 2x then i half the x -value for example?

I don't understand what you're saying.

so if the function instead of$y=\dfrac{3}{x+2}$ was $y=\dfrac{3}{2x}$ would the x-values be halved? i don't think this function $y=\dfrac{3}{2x}$ would touch any axis right?

It would represent and horizontal stretch scale factor $\frac{1}{2}$ or a vertical stetch scale factor $\frac{1}{2}$ (the vertical/horizontal thingies being the same is due to the fact that 1/x is its own inverse function. Cool, right? Anyways, doesn't matter). There still wouldn't be any axis intersections, yes. It is not a shift, just a stretch.

ah right i see thanks for that ^-^