Advertisements
Advertisements
Question
Show that f (x) = | cos x | is a continuous function.
Advertisements
Solution
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
RELATED QUESTIONS
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
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{((kcosx)/(pi-2x)", if" x != pi/2),(3", if" x = pi/2):}` at x = `"pi/2`
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx^2", if" x<= 2),(3", if" x > 2):}` at x = 2
Examine the continuity of the function
\[f\left( x \right) = \left\{ \begin{array}{l}3x - 2, & x \leq 0 \\ x + 1 , & x > 0\end{array}at x = 0 \right.\]
Also sketch the graph of this function.
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
If \[f\left( x \right) = \frac{2x + 3\ \text{ sin }x}{3x + 2\ \text{ sin } x}, x \neq 0\] If f(x) is continuous at x = 0, then find f (0).
Prove that the function \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}\] is everywhere continuous.
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if } x = 2\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x
What happens to a function f (x) at x = a, if
If the function \[f\left( x \right) = \frac{\sin 10x}{x}, x \neq 0\] is continuous at x = 0, find f (0).
Determine the value of the constant 'k' so that function f
The function \[f\left( x \right) = \begin{cases}1 , & \left| x \right| \geq 1 & \\ \frac{1}{n^2} , & \frac{1}{n} < \left| x \right| & < \frac{1}{n - 1}, n = 2, 3, . . . \\ 0 , & x = 0 &\end{cases}\]
If the function f (x) defined by \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos 10x}{x^2} , & x < 0 \\ a , & x = 0 \\ \frac{\sqrt{x}}{\sqrt{625 + \sqrt{x}} - 25}, & x > 0\end{cases}\] then the value of a so that f (x) may be continuous at x = 0, is
Find the values of a and b, if the function f defined by
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
The function f (x) = |cos x| is
Let f (x) = |cos x|. Then,
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
The function f(x) = `"e"^|x|` is ______.
`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.
`lim_("x" -> 0) (1 - "cos" 4 "x")/"x"^2` is equal to ____________.
`lim_("x" -> 0) (1 - "cos x")/"x sin x"` is equal to ____________.
Let f(x) = `{{:(5^(1/x), x < 0),(lambda[x], x ≥ 0):}` and λ ∈ R, then at x = 0
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx + 1",", if x ≤ pi),(cos x",", if x > pi):}` at = `pi`
Find the values of `a` and ` b` such that the function by:
`f(x) = {{:(5",", if x ≤ 2),(ax + b",", if 2 < x < 10),(21",", if x ≥ 10):}`
is a continuous function.
Discuss the continuity of the following function:
f(x) = sin x + cos x
Discuss the continuity of the following function:
f(x) = sin x – cos x
For \[\frac{f}{g}\] to be continuous at \[x=c\], which condition is required?
Which statement about \[\sin x\] and \[\cos x\] is correct?
Let \[g(x)=\sin x\] and \[h(x)=x^2\]. Which expression equals \[(g\circ h)(x)\]?
Why is \[g(x)=1-x+|x|\] continuous?
