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Define a function as a set of ordered pairs.
Define a function as a correspondence between two sets.
What is the fundamental difference between a relation and a function? Is every relation a function?
Let A = {–2, –1, 0, 1, 2} and f : A → Z be a function defined by f(x) = x2 – 2x – 3. Find:
i. range of f i.e. f(A)
ii. pre-images of 6, –3 and 5
If a function f: R → R be defined by
Find: f(1), f(–1), f(0), f(2).
A function f : R → R is defined by f(x) = x2. Determine
i. range of f
ii. {x: f(x) = 4}
iii. {y: f(y) = –1}
Let f: R+→ R, where R+ is the set of all positive real numbers, be such that f(x) = logex. Determine
i. the image set of the domain of f
ii. {x: f(x) = –2}
iii. whether f(xy) = f(x) + f(y) holds.
Write the following relations as sets of ordered pairs and find which of them are functions:
i. {(x, y): y = 3x, x ∈ {1, 2, 3}, y ∈ {3, 6, 9, 12}}
ii. {(x, y): y > x + 1, x = 1, 2 and y = 2, 4, 6}
iii. {(x, y): x + y = 3, x, y ∈ {0, 1, 2, 3}}
Let f : R → R and g : C → C be two functions defined as f(x) = x2 and g(x) = x2. Are they equal functions?
If f, g, h are three functions defined from R to R as follows:
i. f(x) = x2
ii. g(x) = sin x
iii. h(x) = x2 + 1
Find the range of each function.
Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}. Determine which of the following sets are functions from X to Y.
i. f1 = {(1, 1), (2, 11), (3, 1), (4, 15)}
ii. f2 = {(1, 1), (2, 7), (3, 5)}
iii. f3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}
Let A = {12, 13, 14, 15, 16, 17} and f : A → Z be a function given by f(x) = highest prime factor of x. Find range of f.
If f : R → R be defined by f(x) = x2 + 1, then find f-1{17} and f-1{–3}.
Let A = {p, q, r, s} and B = {1, 2, 3}. Which of the following relations from A to B is not a function?
i. R1 = {(p, 1), (q, 2), (r, 1), (s, 2)}
ii. R2 = {(p, 1), (q, 1), (r, 1), (s, 1)}
iii. R3 = {(p, 1), (q, 2), (p, 2), (s, 3)}
iv. R4 = {(p, 2), (q, 3), (r, 2), (s, 2)}
Let A = {9, 10, 11, 12, 13} and let f : A → Z be a function given by f(n) = the highest prime factor of n. Find the range of f.
The function f is defined by
The relation g is defined by
Show that f is a function and g is not a function.
If f(x) = x2, find
Express the function f : X → R given by f(x) = x3 + 1 as set of ordered pairs, where X = {–1, 0, 3, 9, 7}.
If f(x) = x2 – 3x + 4, then find the values of x satisfying the equation f(x) = f(2x + 1).
If f(x) = (x – a)2(x – b)2, find f(a + b).
If, show that x = f(y).
If, show that f[f{f(x)}] = x.
If, show that f[f(x)] = x.
If, find:
i.
ii. f(–2)
iii. f(1)
iv.
v.
If, show that.
If, show that f(tanθ) = sin2θ.
If, then show that
ii.
If, a > 0 and n ∈ N, then prove that f[f(x)] = x for all x.
If for non-zero x,, where a ≠ b, then find f(x).
Find the domain of each of the following real valued functions of real variable:
iii.
Find the domain and range of each of the following real valued functions:
f(x) = |x – 1|
f(x) = –|x|
Find f + g, f – g, cf (c ∈ R, c ≠ 0), fg, 1/f and f/g in each of the following:
f(x) = x3 + 1 and g(x) = x + 1
and
Let f(x) = 2x + 5 and g(x) = x2 + x. Describe
i. f + g
ii. f – g
iii. fg
Find the domain in each case.
If f(x) be defined on [–2, 2] and is given by and g(x) = f(|x|) + |f(x)|. Find g(x).
Let f, g be two real functions defined by and . Then, describe each of the following functions.
ii. g – f
vi.
vii. f2 + 7f
viii.
If f(x) = loge(1 – x) and g(x) = [x], then determine each of the following functions:
ii. fg
Also, find (f + g)(–1), (fg)(0), and .
If f, g, h are real functions defined by, and h(x) = 2x2 – 3, then find the values of (2f + g – h)(1) and (2f + g – h)(0).
The function f is defined by . Draw the graph of f(x).
Let f, g : R → R be defined, respectively by f(x) = x + 1 and g(x) = 2x – 3. Find f + g, f – g and.
Let f : [0, ∞) → R and g : R → R be defined by and g(x) = x. Find f + g, f – g, fg and
Let f(x) = x2 and g(x) = 2x + 1 be two real functions. Find (f + g)(x), (f – g)(x), (fg)(x) and .
Write the range of the real function f(x) = |x|.
If f is a real function satisfying for all x ∈ R – {0}, then write the expression for f (x).
Write the range of the function f (x) = sin [x] where
If f(x) = cos2[π2]x + cos [-π2] x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).
Write the range of the function f (x) = cos [x], where
Write the range of the function f(x) = ex – [x], X ∈R.
LetThen write the value of α satisfying f(x)) = x for allx ≠ - 1.
If then write the value of
Write the domain and range of the function
If f(x) = 4x – x2, x ∈ R, then write the value of f(a + 1) – f(a – 1).
If f, g, h are real functions given by f(x) = x2, g(x) = tan x and h(x) = loge x, then write the value of (hogof)
Write the domain and range of function f(x) given by
Write the domain and range of
Let A and B be two sets such that n(A) = p and n(B) = q, write the number of functions from A to B.
Let f and g be two functions given by
f = {(2, 4), (5, 6), (8, -1), (10, -3)} and g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, -5)}.
Find the domain of f + g.
Find the set of values of x for which the functions f(x) = 3x2 – 1 and g(x) = 3 + x are equal.
Let f and g be two real functions given by
f = {(0, 1), (2, 0), (3, -4), (4, 2), (5, 1)} and g = {(1, 0), (2, 2), (3, -1), (4, 4), (5, 3)}.
Find the domain of fg.
Mark the correct alternative in the following:
Let A = {1, 2, 3}, B = {2, 3, 4}, then which of the following is a function from A to B?
If f : Q → Q is defined as f(x) = x2, then f-1 (9)s is equal to
Which one of the following is not a function?
If f(x) = cos (log x), then has the value
Let f(x) = |x – 1|. Then,
The range of f(x) = cos [x], for -π/2< x <π/2 is
Which of the following are functions?
If and then f(g(x) is equal to
If A = {1, 2, 3}, B = {x, y}, then the number of functions that can be defined from A into B is
If then is equal to
If f(x) = cos (log x), then value of is
If then f(x + y) f(x – y) is equals to
If then f(2) is equal to
Let f : R → r be defined by f(x) = 2x + |x|. Then f(2x) + f(-x) – f(x) =
The range of the function is
If x ≠ 1 and is a real function, the f(f(f(2)) is
If f(x) = cos (loge x), then is equal to
Let and h(x) = f(x) g(x). Then, h(x) = 1 for
If for x ∈ R, then f (2002) =
The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =
Let A = {x ∈R : x ≠ 0, -4 ≤ x ≤ 4} and f : A → R be defined by for x ∈ A. Then A is
If f : R → R and g : R → R are defined by f(x) = 2x + 3 and g(x) = x2 + 7, then the values of x such that g(f (x)) = 8 are
If f : [-2, 2] → R is defined by then
{x [-2, 2] : x ≤ 0 and f(|x|) = x} =
If and then k =
If f is a real valued function given by and α, β are roots of Then,
If and α, β are the roots of Then,
If for all non-zero x, then f(x) =
If f : R → R be given by for all x ∈ R. Then,
If f(x) = sin [π2] x + sin [-π2] x, where [x] denotes the greatest integer less than or equal to x, then
The domain of the function is
The domain of definition of is
The domain of definition of the function is
The domain of definition of the functionis
The range of the function f(x) = |x – 1| is
Let Then, which of the following is correct?
If [x]2 – 5[x] + 6 = 0, where [∙] denotes the greatest integer function, then
The range of is
