Advertisements
Advertisements
प्रश्न
Show that f (x) = | cos x | is a continuous function.
Advertisements
उत्तर
The given function is `f(x)=|cos x|`
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g o h, where `g(x)=|x| and h(x)=cos x`
`[∵(goh)(x)=g(h(x))=g(cos x)=|cos x|=f(x)]`
It has to be first proved that `g(x)=|x| and h(x)=cos x` are continuous functions.
`g(x)=|x| " can be written as " `
`g(x)=[[-x,if x≤ 0],[x,if x ≥ 0]]`
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
`if c < 0 " then " g (c)= -c and lim\_(x->c)(-x)=-c`
`∴ lim_(x->c)g(x)=g(c)`
So, g is continuous at all points x < 0.
Case II:
`" if c < 0 then " g (c)= -c and lim\_(x->c)(-x)=-c`
`∴ lim_(x->c)g(x)=g(c)`
So, g is continuous at all points x > 0.
Case III:
`if c = 0 , " then " g(c)=g(0)=0`
`lim_(x->0^-)g(x)=lim_(x->0^-)(-x)=0`
`lim_(x->0^+)g(x)=lim_(x->0^+)(x)=g(0)`
`∴lim_(x->0^+)g(x)=lim_(x->0^+)(x)=g(0)`
So, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points.
Now, h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number.
Put x = c + h
If x → c, then h → 0
h (c) = cos c
`lim_(x->0)h(x)=lim_(x->0) cos x`
`=lim_(k->0) cos (c+h)`
`=lim_(k->0)[cos c cos h-sin c sin h]`
`=lim_(k->0)cos c cos 0 - sin c sin 0`
`= cos c xx1 - sin cxx0`
`= cos c`
`lim_(x->c)h(x)=h(c)`
So, h (x) = cos x is a continuous function.
It is known that for real valued functions g and h,such that (g o h) is defined at x = c, if g is continuous at x = c and if f is continuous at g (c), then (f o g) is continuous at x = c.
Therefore, `f(x)=(goh)(x)=g(h(x))=g(cos x)=|cos x|` is a continuous function.
APPEARS IN
संबंधित प्रश्न
If f (x) is continuous on [–4, 2] defined as
f (x) = 6b – 3ax, for -4 ≤ x < –2
= 4x + 1, for –2 ≤ x ≤ 2
Show that a + b =`-7/6`
Find the relationship between a and b so that the function f defined by f(x) = `{(ax + 1", if" x<= 3),(bx + 3", if" x > 3):}` is continuous at x = 3.
Discuss the continuity of the following function:
f(x) = sin x × cos x
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
Find the values of a and b such that the function defined by f(x) = `{(5", if" x <= 2),(ax +b", if" 2 < x < 10),(21", if" x >= 10):}` is a continuous function.
Extend the definition of the following by continuity
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\pi x}{2}, \text{ if } & x \neq 1 \\ k , if & x = 1\end{cases}\] at x = 1at x = 1
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & \text{ if } x \neq 0 \\ 3k , & \text{ if } x = 0\end{cases}\]
In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if } x < 0 \\ \cos 2x , & \text{ if } x \geq 0\end{cases}\]
Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.
What happens to a function f (x) at x = a, if
If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then write the value of k.
If \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\] is continuous at x = 0, find k.
Determine the value of the constant 'k' so that function f
Let \[f\left( x \right) = \left\{ \begin{array}\\ \frac{x - 4}{\left| x - 4 \right|} + a, & x < 4 \\ a + b , & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, & x > 4\end{array} . \right.\]Then, f (x) is continuous at x = 4 when
\[f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)\] is continuous, is given by
If the function \[f\left( x \right) = \frac{2x - \sin^{- 1} x}{2x + \tan^{- 1} x}\] is continuous at each point of its domain, then the value of f (0) is
Find the values of a and b, if the function f defined by
The function f (x) = 1 + |cos x| is
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]
then at x = 0, f (x) is
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
Let f(x) = |sin x|. Then ______.
`lim_("x" -> 0) (1 - "cos" 4 "x")/"x"^2` is equal to ____________.
A real value of x satisfies `((3 - 4ix)/(3 + 4ix))` = α – iβ (α, β ∈ R), if α2 + β2 is equal to
If `f(x) = {{:(-x^2",", "when" x ≤ 0),(5x - 4",", "when" 0 < x ≤ 1),(4x^2 - 3x",", "when" 1 < x < 2),(3x + 4",", "when" x ≥ 2):}`, then
The function f(x) = 5x – 3 is continuous at x =
The function f(x) = x |x| is ______.
Which concept is fundamental in determining continuity of a function at a point?
If \[f\] and \[g\] are continuous at \[x=c\], which collection is continuous at \[x=c\]?
If \[g\] is continuous and \[g(x)\neq 0\], which function is continuous?
For \[f(x)=\frac{p(x)}{q(x)}\], where \[p(x)\] and \[q(x)\] are polynomial functions, where is \[f\] continuous?
At which values is \[\tan x=\dfrac{\sin x}{\cos x}\] not continuous?
For \[f(x)=|1-x+|x||\], which functions give the representation \[f(x)=h(g(x))\]?
