Question

write the equation of a non - linear function such that $lim_{x
ightarrow - 1}\frac{f(x)-f(-1)}{x + 1}=-1$. $f(x)=$

Explanation:

Step1: Recall the definition of the derivative

The limit $\lim_{x
ightarrow - 1}\frac{f(x)-f(-1)}{x + 1}$ is the definition of the derivative of the function $y = f(x)$ at $x=-1$, i.e., $f'(-1)$. So we know that $f'(-1)=-1$.

Step2: Choose a non - linear function

A simple non - linear function is a quadratic function of the form $f(x)=ax^{2}+bx + c$. The derivative of $f(x)$ using the power rule $(x^{n})'=nx^{n - 1}$ is $f'(x)=2ax + b$.

Step3: Substitute $x=-1$ into the derivative

We want $f'(-1)=-1$. Substituting $x = - 1$ into $f'(x)=2ax + b$ gives $f'(-1)=-2a + b$. Let $a = 1$, then $-2a + b=-1$ becomes $-2 + b=-1$, so $b = 1$. We can choose $c = 0$ for simplicity.
So a non - linear function $f(x)=x^{2}+x$ satisfies the condition.

Answer:

$f(x)=x^{2}+x$