Show that the function defined by f(x) = |cos x| is a continuous function

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 = goh, where g(x) = |x| and h(x) = cosx

[Therefore (goh)(x) = g(h(x)) = g(cosx) = |cosx| = f(x)]

It has to be proved first 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) 

It is evident that g is defined for every real number.

Let c be a real number.

Case I:

If c < 0, then g(c) = −c 

lim_x→c g(x) = lim_x→c (−x) = −c

⇒ lim_x→c g(x) = g(c) 

Therefore, g is continuous at all points x, such that x < 0.

Case II:

If c > 0, then g(c) = c

lim_x→c g(x) = lim_x→c (x) = c

⇒ lim_x→c g(x) = g(c) 

Therefore, g is continuous at all points x, such that 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) = 0

⇒ lim_x→0⁻ g(x) = lim_x→0⁺ (x) = g(0)

Therefore, g is continuous at all x = 0.

From the above three observations, it can be concluded that g is continuous at all points.

Let 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→c h(x)

= lim_x→c cos x = lim_h→0 cos(c + h)

= lim_h→0 [cos c cos h − sin c sin h]

= lim_h→0 (cos c cos h) - lim_h→0 (sin c sin h)

= cos c cos 0 − sin c sin 0

= cos c (1) − sin c (0)

= cos ⁡c

⇒ lim_x→c h(x) = h(c)

Therefore, h(x) = cos x is a continuous function.

It is known that for real valued functions g and h, such that (goh) is defined at c,

if g is continuous at c and if f is continuous at g(c), then (fog) is continuous at c.

Therefore, f(x) = (goh)(x) = g(h(x)) = g(cosx) = |cosx| is a continuous function

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 32

Show that the function defined by f(x) = |cosx| is a continuous function

Summary:

Hence we can conclude that f(x) = (goh)(x) = g(h(x)) = g(cosx) = |cosx| is a continuous function