Advertisements
Advertisements
Question
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 (fh).
Advertisements
Solution
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, (fh) (x)=f (x)h (x) = (sin x) (cos x) =`1/2`sin (2x)
Domain of fh is R.
Since range of sin x is [-1,1],
−1 ≤ sin 2x ≤ 1
⇒ ` (-1)/2 ≤ sin x/2 ≤ 1/2`
Range of fh = `[(-1)/2 ","1/2]`
So, (fh) : R →`[(-1)/2 ","1/2]`
Clearly, range of fh is a subset of g.
⇒ go (fh) : R → R
⇒ domains of fog and go (fh) are the same .
So, (fog) (x)=f (g (x)) = f (2x) = sin (2x)
and ( go (fh)) (x) = g ((fh) (x)) = g (sinx cos x) = 2sin x cos x = sin (2x)
⇒ (fog) (x) = ( go(fh)) (x), ∀x ∈ R
Hence, fog = go (fh)
APPEARS IN
RELATED QUESTIONS
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
Let \[f : \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \to R\] be a function defined by f(x) = cos [x]. Write range (f).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
Let f be an injective map with domain {x, y, z} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.
\[f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .\]
The value of
\[f^{- 1} \left( 1 \right)\] is
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x2
Which of the following functions from Z into Z are bijections?
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Which of the following functions from Z into Z is bijective?
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: R → R be defined by f(x) = x2 is:
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' is
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
A function is a rule that assigns each element of one set to exactly how many elements of another set?
For \(f(x)=x^2+x\), if \(y=1\), which two domain elements have the same image?
In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:
For \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=1+x^2\), which statement is correct?
Many students choosing the same favourite subject resembles which type of function?
Which condition represents a many-one function?
