\(f(x)=\left\{\begin{array}{cc}\frac{x-4}{|x-4|}+a, & x 4\end{array}\right.\) If \(f(x)\) given above is…
\(f(x)=\left\{\begin{array}{cc}\frac{x-4}{|x-4|}+a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|}+b, & x>4\end{array}\right.\)
If \(f(x)\) given above is continuous at \(x=4\), then find the values of ' \(a\) ' and ' \(b\) '.
\(a=1, b=-1\)
\(a=-1, b=1\)
\(a=1, b=1\)
\(a=-1, b=-1\)
Solution
Given function
\(\begin{aligned}
f(x) & =\left[\begin{array}{cc}
\frac{x-4}{|x-4|}+a, & x < 4 \\
a+b, & x=4 \\
\frac{x-4}{|x-4|}+b, & x > 4
\end{array}\right. \\
& =\left[\begin{array}{cl}
-1+a, & x < 4 \\
a+b, & x=4 \\
1+b, & x > 4
\end{array}\right.
\end{aligned}\)
\(\because\) Function \(f\) is continuous at \(x=4\), so
LHL (at \(x=4)=f(4)=\) RHL (at \(x=4\))
\(\Rightarrow-1+a=a+b=1+b \Rightarrow a=1\) and \(b=-1\)
Hence, option (a) is correct.