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failure function tests
1 FAILURE FUNCTION
An abstract definition is:
Let $\phi(x)$ be a function of $x$. Then, $x=\psi(x_{0})$ is a failure function if the values of $x$ generated by $\psi(x_{0})$, when substituted in $\phi(x)$, generate only failures in accordance with our definition of a failure. Here $x_{0}$ is a specific value of $x$.
1.1 Examples

(i) Let the mother function be a polynomial in x (coeffficients belong to $\mathcal{Z}$ ), say $\phi(x)$. Let our definition of a failure be a composite number. Then, $x=\psi(x_{0})=x_{0}+k(\phi(x_{0}))$ is a failure function because the values of x generated by $\phi(x_{0})$, when substituted in $\phi(x)$ , generate only failures.

(ii) Let the mother function be an exponential function, say $\phi(x)=a^{x}+c$. Then $x=\psi(x_{0})=x_{0}+k.Eulerphi(\phi(x_{0}))$ is a failure function since the values of x generated by $\psi(x_{0})$, when substituted in the mother function, generate only failures.

(iii) Let our definition of a failure be a nonCarmichael number. Let the mother function $\phi(x)$ be $2^{n}+49$. Then, $n=5+6k$ is its failure function $\psi(x)$.
1.2 Note
Here too our definition of a failure is a composite number and k belongs to N.
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