Advertisements
Advertisements
Question
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f · g)oh = (foh)·(goh)
Advertisements
Solution
To prove:
(f + g)oh = foh + goh
Consider:
((f + g)oh)(x)
= (f + g)(h(x))
= f(h(x)) + g(h(x))
= (foh)(x) + (goh)(x)
= {(foh) + (goh)}(x)
∴ ((f + g)oh)(x) = {(foh) + (goh)}(x) ∀ x ∈ R
Hence, (f + g)oh = foh + goh.
To prove:
(f · g)oh = (foh)·(goh)
Consider:
((f · g)oh)(x)
= (f · g)(h(x))
= f(h(x))·g(h(x))
= (foh)(x)·(goh)(x)
= {(foh)·(goh)}(x)
∴ ((f · g)oh)(x) = {(foh)·(goh)}(x) ∀ x ∈ R
Hence, (f · g) oh = (foh)·(goh).
RELATED QUESTIONS
Let f : W → W be defined as
`f(n)={(n-1, " if n is odd"),(n+1, "if n is even") :}`
Show that f is invertible a nd find the inverse of f. Here, W is the set of all whole
numbers.
Find gof and fog, if f(x) = 8x3 and `g(x) = x^(1/3)`.
State with reason whether following functions have inverse
f: {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
State with reason whether following functions have inverse
g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
State with reason whether following functions have inverse
h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f−1 and show that (f−1)−1 = f.
Let f: X → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f−1)−1 = f.
If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.
Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.
If f: R → R is defined by f(x) = x2 − 3x + 2, find f(f(x)).
Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`
Hence Find
1) `f^(-1)(10)`
2) y if `f^(-1) (y) = 4/3`
where R+ is the set of all non-negative real numbers.
Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = `x/(x^2 + 1)` Then gof is ______.
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.
Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.
Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.
The composition of functions is commutative.
The composition of functions is associative.
If f(x) = (ax2 + b)3, then the function g such that f(g(x)) = g(f(x)) is given by ____________.
Let f : N → R : f(x) = `((2"x"−1))/2` and g : Q → R : g(x) = x + 2 be two functions. Then, (gof) `(3/2)` is ____________.
If f(x) = `(3"x" + 2)/(5"x" - 3)` then (fof)(x) is ____________.
Let f : R → R be the functions defined by f(x) = x3 + 5. Then f-1(x) is ____________.
Let f : R – `{3/5}`→ R be defined by f(x) = `(3"x" + 2)/(5"x" - 3)` Then ____________.
If f(x) = (ax2 – b)3, then the function g such that f{g(x)} = g{f(x)} is given by ____________.
Which one of the following functions is not invertible?
If f is an invertible function defined as f(x) `= (3"x" - 4)/5,` then f-1(x) is ____________.
The domain of definition of f(x) = log x2 – x + 1) (2x2 – 7x + 9) is:-
Domain of the function defined by `f(x) = 1/sqrt(sin^2 - x) log_10 (cos^-1 x)` is:-
Let A = `{3/5}` and B = `{7/5}` Let f: A → B: f(x) = `(7x + 4)/(5x - 3)` and g:B → A: g(y) = `(3y + 4)/(5y - 7)` then (gof) is equal to
If f: A → B and G B → C are one – one, then g of A → C is
If f(x) = [4 – (x – 7)3]1/5 is a real invertible function, then find f–1(x).
Let `f : R {(-1)/3} → R - {0}` be defined as `f(x) = 5/(3x + 1)` is invertible. Find f–1(x).
