Question

Let $f\left(x\right)=\{\begin{array}{ccc}x+1:& x\le 1& \\ 3-a{x}^2:& x>1& \end{array}$.

Find the value of $a$ if $f$ is continuous at $x=1$.

Answer 1

Given the function is $f\left(x\right)=\{\begin{array}{ccc}x+1:& x\le 1& \\ 3-a{x}^2:& x>1& \end{array}$.

Also given the function $f\left(x\right)$ to be continuous at $x=1$.

Then,

$\underset{x\to 1+}{lim}f\left(x\right)=\underset{x\to 1-}{lim}f\left(x\right)=f\left(1\right)$

or, $3-a=2=2$ [ Using the definition of the function]

or, $a=3-2$

or, $a=1$.