If $ f(x) = \frac{x}{2} -1 $ then on the interval $ [0,\pi] $
- $tan [f(x)]$ and $ \frac{1}{f(x)} $ are both continuous
- $tan [f(x)]$ and $ \frac{1}{f(x)} $ are discontinuous
- $tan [f(x)]$ is continuous but $ \frac{1}{f(x)} $ is not continuous
- $tan[f(x)]$ is not continuous but $ \frac{1}{f(x)} $ is continuous
The Correct Option is C
Solution and Explanation
$\therefore tan(f(x)) = tan (\frac{x}{2} - 1)$
and $\frac{1}{f(x)} = \frac{1}{\frac{x}{2} - 1} = \frac{2}{x -2}$
By using graph transformation method we can draw the graph of
$tan(f(x))$ and $\frac{1}{f(x)}$ as follow :
Graph of $tan\,f(x)$
Graph at $tan[f(x)]$ in $x \in (-\pi + 2 , \pi + 2)$
Graph of $\frac{1}{f(x)}$
From the above graphs of $tan(f(x))$ and $\frac{1}{f(x)}$ is continuous in $ x \in [0, \pi]$ but $\frac{1}{f(x)}$ is not continuous in $x \in [0, \pi]$
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.