Question
Show that the function $\text{f(x)}=\big|\sin\text{x}+\cos\text{x}|$ is continuous at $\text{x}=\pi.$
✓
Answer
Consider, $\text{f(x)}=\big|\sin\text{x}+\cos\text{x}\big|\text{ at x}=\pi$
Let $\text{g(x)}=\sin\text{x}+\cos\text{x}$
And $\text{h(x)}=|\text{x}|$
$\therefore\ \text{hog (x)}=\text{h}[\text{g (x)}]$
$=\text{h }(\sin\text{x}+\cos\text{x})$
$=|\sin\text{x}+\cos\text{x}|$
Since, g(x) and h(x) are continuous functions and f(x) is a composite functions.
We know that composite functions of two continuous functions is also a continuous function.
Hence, $\text{f(x)}=|\sin\text{x}+\cos\text{x}|$ is a continuous function everywhere,
So, f(x) is continuous at $\text{x}=\pi.$
Let $\text{g(x)}=\sin\text{x}+\cos\text{x}$
And $\text{h(x)}=|\text{x}|$
$\therefore\ \text{hog (x)}=\text{h}[\text{g (x)}]$
$=\text{h }(\sin\text{x}+\cos\text{x})$
$=|\sin\text{x}+\cos\text{x}|$
Since, g(x) and h(x) are continuous functions and f(x) is a composite functions.
We know that composite functions of two continuous functions is also a continuous function.
Hence, $\text{f(x)}=|\sin\text{x}+\cos\text{x}|$ is a continuous function everywhere,
So, f(x) is continuous at $\text{x}=\pi.$
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