Question

f(x)= (x+ radical x) / (radical x)

a) Explain why the function f has one or more holes in its graph, and state the x values at which those holes occur.

b) Find a function g whose graph is identical to that of f, but without the holes

Answer 1

\[f(x) = x + \frac{\sqrt{x}}{\sqrt{x}}\]
I'm not sure it really has a hole. My understanding is that a hole requires the function to be defined on both sides of a hole,
\[\sqrt{0} = 0\] and \[\frac{anything}{0} = undefined\]
So clearly x = 0 is not defined. However x < 0 is not defined either for any real number as sqrt of a negative number is a complex number.

So I would say that f(x) is only defined for x > 0. And when x > 0 then sqrt(x)/sqrt(x) = 1

Maybe I'm missing something.

Answer 2

hmm ok thanks for your help!

Answer 3

the equation is actually \[\frac{ x+\sqrt{x} }{ \sqrt{x} }\]

Answer 4

hmm, I must have read it wrong.

\[\frac{x}{\sqrt{x}} + \frac{\sqrt{x}}{\sqrt{x}} => \frac{x \sqrt{x}}{\sqrt{x}\sqrt{x}} + 1 => \sqrt{x} + 1 \]

I still think the function is only defined for x > 0.

Answer 5

ok thanks! i ended up getting that too :)