$($CS$417/517)$ Homework $3$

$1.(10$ points$)$ Let f$($x$)=$

$(1+$ x$^8)1^/41$

a$)$ Explain the difficulty of computing f$($x$)$ for a small value of $|$x$|$ and show how it can be circumvented.

b$)$ Compute $($condf f$)($x$)$ and discuss the conditioning of f$($x$)$ for small $|$x$|.$

c$)$ How can the answers to $($a$)$ and $($b$)$ be reconciled $($i$.$e$.,$ how does the cancellation error analysis impact the condition number$)?$