Question

Find the relationship between $a$ and $b$ so that the function $f$ defined by,
f\left( x \right)=\left\\{ \begin{matrix} ax+1 & \text{if }x\le 3 \\\ bx+3 & \text{if }x>3 \\\ \end{matrix} \right.
Is continuous at $x=3$

Answer 1

In this question we are given a function which has two values when the value is above, lesser than or equal to $3$. We know that the function is continuous at $x=3$ which means that the left-hand limit and the right-hand limit will be the same when the value of $x=3$ is substituted in the function. We will equate both the left-hand limit and the right-hand limit and simplify the terms to get the relation between the variables $a$ and $b$.

Complete step by step solution:
We have f\left( x \right)=\left\\{ \begin{matrix} ax+1 & \text{if }x\le 3 \\\ bx+3 & \text{if }x>3 \\\ \end{matrix} \right.
We know that the function is continuous at $x=3$.
This indicates that $\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)=f\left( 3 \right)$.
Therefore, we can equate both the equations by substituting the value of $x=3$ in $f\left( x \right)$.
On substituting and equating, we get:
$\Rightarrow a\times 3+1=b\times 3+3$
On multiplying the terms in the left-hand side and the right-hand side, we get:
$\Rightarrow 3a+1=3b+3$
On transferring the term $1$ from the left-hand side to the right-hand side, we get:
$\Rightarrow 3a=3b+3-1$
On simplifying the right-hand side, we get:
$\Rightarrow 3a=3b+2$
On transferring the term $3b$ from the right-hand side to the left-hand side, we get:
$\Rightarrow 3a-3b=2$
Now we can see that $3$ is common in the left-hand side therefore, on taking it out as common, we get:
$\Rightarrow 3\left( a-b \right)=2$
On transferring the term $3$ from the left-hand side to the right-hand side, we get:
$\Rightarrow a-b=\dfrac{2}{3}$, which is the relation between the terms $a$ and $b$ which satisfies the given function.

Note: It is to be remembered that a function is said to be continuous if it is continuous on all the points in its given domain. In the given question we can see that the function is not continuous at other values except $x=3$ because the function changes when $x\le 3$ or when $x<3$.