Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (f h).

Asked by Aaryan | 1 year ago |  24

##### Solution :-

Given that f(x) = sin x, g (x) = 2x and h (x) = cos x

We know that f: R→ [−1, 1] and g: R→ R

Clearly, the range of g is a subset of the domain of f.

fog: R → R

Now, (f h) (x) = f (x) h (x) = (sin x) (cos x) = $$\dfrac{ 1}{2}$$ sin (2x)

Domain of f h is R.

Since range of sin x is [-1, 1], −1 ≤ sin 2x ≤ 1

$$\dfrac{ -1}{2}$$ ≤ sin $$\dfrac{ x}{2}$$ ≤ $$\dfrac{ 1}{2}$$

Range of f h = [$$\dfrac{ -1}{2}$$, $$\dfrac{ 1}{2}$$]

So, (f h): R → [$$\dfrac{ -1}{2}$$,$$\dfrac{ 1}{2}$$]

Clearly, range of f h is a subset of g.

⇒ go (f h): R → R

⇒ Domains of fog and go (f h) are the same.

So, (fog) (x) = f (g (x))

= f (2x)

= sin (2x)

And (go (f h)) (x) = g ((f(x). h(x))

= g (sin x cos x)

= 2sin x cos x

= sin (2x)

⇒ (fog) (x) = (go (f h)) (x), ∀x ∈ R

Hence, fog = go (f h)

Answered by Aaryan | 1 year ago

### Related Questions

#### Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Let A be the set of first five natural and let R be a relation on A defined as follows: (x, y) R x ≤ y

Express R and R-1 as sets of ordered pairs. Determine also

(i) the domain of R‑1

(ii) The Range of R.

#### A function f: R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

A function f: R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).

#### If f: R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1.

If f: R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1 (24) and f−1 (5).

Consider f: R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with f-1(x) = $$\dfrac{\sqrt{(x+6)-1}}{3}$$
If f(x) = $$\dfrac{ (4x + 3)}{(6x – 4)}$$, x ≠ ($$\dfrac{2}{3}$$) show that fof(x) = x, for all x ≠ ($$\dfrac{2}{3}$$). What is the inverse of f?