Question

please let f(x)= 1/sqroot x-3 . evaluate the function at ( f(x) - f(4) ) / x-4 . simplify your answer.

Answer 1

\[f(x)=\frac{1}{\sqrt{x-3}}\]
\[f(4)=\frac{1}{\sqrt{4-3}}=1\]
\[\frac{f(x)-f(4)}{x-4}=\frac{\sqrt{x-3}-1}{x-4}\]

Answer 2

now as for "simplifying" i believe what they really mean is rationalize the numerator and cancel

Answer 3

multiply top and bottom by
\[\sqrt{x-3}+1\] so get
\[\frac{\sqrt{x-3}-1}{x-4}\times \frac{\sqrt{x-3}+1}{\sqrt{x-3}+1}\]
\[\frac{x-3-1}{(x-4)(\sqrt{x-3}+1)}\]
\[\frac{x-4}{(x-4)(\sqrt{x-3}+1)}=\frac{1}{\sqrt{x-3}+1}\]