Continuity and Differentiability — Maths STD 12 Science — Question

f(x) has a real and finite value for all $x \in R.$

$\therefore$ Domain of f(x) is R.

Let g(x) = cos x and $h\left( x \right) = \left| x \right|$ 

Since g(x) and h(x) being cosine function and modulus function are continuous for all real x 

Now, $\left( {goh} \right)x = g\left\{ {h\left( x \right)} \right\} = g\left( {\left| x \right|} \right) = \cos \left| x \right|$ being the composite function of two continuous functions is continuous, but not equal to f(x) 

Again, $\left( {hog} \right)x = h\left\{ {g\left( x \right)} \right\} = h\left( {\cos x} \right) = \left| {\cos x} \right| = f\left( x \right)$[Using eq. (i)]

Therefore, $f\left( x \right) = \left| {\cos x} \right| = \left( {hog} \right)x$ being the composite function of two continuous functions is continuous.