Question

If $f\left(x\right)=\left\{\begin{array}{cc}ax+1,& \mathrm{if}x\ge 1\\ x+2,& \mathrm{if}x<1\end{array}\right.$is continuous, then 'a' should be equal to __________.

Answer 1

It is given that, the function $f\left(x\right)=\left\{\begin{array}{cc}ax+1,& \mathrm{if}x\ge 1\\ x+2,& \mathrm{if}x<1\end{array}\right.$is continuous.

So, the function f(x) is continuous at x = 1.

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

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

Now,

f(1) = a × 1 + 1 = a + 1 .....(2)

$\underset{x\to 1^{+}}{\lim }f\left(x\right)=\underset{x\to 1}{\lim }\left(ax+1\right)=a\times 1+1=a+1$ .....(3)

$\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1}{\lim }\left(x+2\right)=1+2=3$ .....(4)

From (1), (2), (3) and (4), we have

a + 1 = 3 = a + 1

⇒ a = 3 − 1 = 2

Thus, the value of a is 2.


If $f\left(x\right)=\left\{\begin{array}{cc}ax+1,& \mathrm{if}x\ge 1\\ x+2,& \mathrm{if}x<1\end{array}\right.$is continuous, then 'a' should be equal to ___2___.