Advertisements
Advertisements
Question
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Solution
Here f(x) = |x| + x which can be redefined as
f(x) = `{(2x, "if" x ≥ 0),(0, "if" x < 0):}`
Similarly, the function g defined by g(x) = |x| – x may be redefined as
g(x) = `{(0, "if" x ≥ 0),(-2x, "if" x < 0):}`
Therefore, g o f gets defined as:
For x ≥ 0, (g o f) (x) = g (f(x) = g (2x) = 0
and for x < 0, (g o f) (x) = g (f(x) = g (0) = 0.
Consequently, we have (g o f) (x) = 0, ∀ x ∈ R.
Similarly, f o g gets defined as:
For x ≥ 0, (f o g) (x) = f (g(x) = f(0) = 0,
and for x < 0, (f o g) (x) = f (g(x)) = f(–2x) = – 4x.
i.e. (f o g) (x) = `{(0, x > 0),(-4x, x < 0):}`
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f: R → R given by f(x) = x2
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Show that function f: R `rightarrow` {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [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
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
Let f: R → R defined by f(x) = 3x. Choose the correct answer
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
Find the domain of sin–1 (x2 – 4).
