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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 function? Is every relation a function?
Let X = {1, 2, 3, 4,}, Y = {1, 5, 9, 11, 15, 16} and F = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}.
Are the following true?
(i) F is a relation from X to Y (ii) F is a function from X to Y. Justify your answer in following true?
Let X = {-1, 0, 3, 7, 9} and f : X → R : f(x) x3 + 1. Express the function f as set of ordered pairs.
Let A = {–1, 0, 1, 2} and B = {2, 3, 4, 5}. Find which of the following are function from A to B. Give reason.
(i) f = {(–1, 2), (-1, 3), (0, 4), 1,5)}
(ii) g = {(0, 2), (1, 3), (2, 4)}
(iii) h = {(-1, 2), (0, 3), (1, 4), (2, 5)}
Let A = {1, 2} and B = {2, 4, 6}. Let f = {(x, y) : x ϵ A, y ϵ B and y > 2x + 1}. Write f as a set of ordered pairs. Show that f is a relation but not a function from A to B.
Let A = {0, 1, 2} and B = {3, 5, 7, 9}. Let f = {(x, y) : x ϵ A, y ϵ B and y = 2x + 3}. Write f as a set of ordered pairs. Show that f is function from A to B. Find dom (f) and range (f).
Let A = {2, 3, 5, 7} and B = {3, 5, 9, 13, 15}. Let f = {(x, y) : x ϵ A, y ϵ B and y = 2x – 1}. Write f in roster form. Show that f is a function from A to B. Find the domain and range of f.
Let g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)}. Is g a function? If yes, its domain range. If no, give reason.
Let f = {(0, -5), (1, -2), (3, 4), (4, 7)} be a linear function from Z into Z. Write an expression for f.
If f (x) = x2, find the value of .
If f(x) = x2, find the value of .
Let X = {12, 13, 14, 15, 16, 17} and f : A → Z : f(x) = highest prime factor of x. Find range (f)
Let R+ be the set of all positive real numbers. Let f : R+→ R : f(x) = logex. Find
(i) range (f)
(ii) {x : x ϵ R+ and f(x) = -2}.
(iii) Find out whether f(x + y) = f(x). f(y) for all x, y ϵ R.
Let f : R → R : f(x) =2x. Find
(ii) {x : f(x) = 1}.
Let f : R → R : f(x) =x2 and g : C → C: g(x) =x2, where C is the set of all complex numbers. Show that f ≠ g.
f, g and h are three functions defined from R to R as following:
(i) f(x) = x2
(ii) g(x) = x2 + 1
(iii) h(x) = sin x
That, find the range of each function.
Let f : R → R : f(x) = x2 + 1. Find
(i) f–1 {10}
(ii) f–1 {–3}.
The function is the formula to convert x °C to Fahrenheit units. Find
(i) F(0),
(ii) F(–10),
(iii) The value of x when f(x) = 212.
Interpret the result in each case.
If f(x) = x2 – 3x + 4 and f(x) = f(2x + 1), find the values of x.
If then show that
(i)
(ii)
If then show that f{f(x)} = x.
If f(x) = and then prove that f{(x)} = , when it is given that .
If then show that f [f{f(x)}] = x.
If then show that f(tanθ) = sin 2θ.
If , prove that x = f(y).
Find the domain of each of the following real function.
(iii)
(iv)
Find the domain and the range of each of the following real function:
Find the domain and the range of each of the following real function: f(x) =
Find the domain and the range of each of the following real function: f(x) = 1 – |x – 2|
Consider the real function f: R → R: f(x) = x + 5 for all
x ϵ R. Find its domain and range. Draw the graph of this function.
Consider the function f: R → R, defined by
Write its domain and range. Also, draw the graph of f(x).
Find the domain and the range of the square root function,
f: R+ U {0} → R f(x) = for all non-negative real numbers.
Also, draw its graph.
Find the domain and the range of the cube root function,
f: R → R: f(x) = x1/3 for all x ϵ R. Also, draw its graph.
Let f : R → R : f(x) = x + 1 and g : R → R : g(x) = 2x – 3.Find
(i) (f + g) (x)
(ii) (f – g) (x)
(iii) (fg) (x)
(iv)(f/g) (x)
Let f : R → R : f(x) = 2x + 5 and g : R → R : g(x) = x2 + x.
Find
(iv) (f/g)(x)
. Let f: R → R: f(x) = x3 + 1 and g: R → R: g(x) = (x + 1). Find:
(iii) (1/f) (x)
(iv) (f/g) (x)
. Let f: R → R; f(x)= ,where c is a constant. Find
(i) (cf) (x)
(ii) (c2 f) (x)
. Let f:(2, ∞) → R: f(x) = and g: (2, ∞) → R: g(x) = Find:
(ii) (f - g) (x)
Find the set of values for which the function f(x) = 1 – 3x and g(x) = 2x2 – 1 are equal.
Find the set of values for which the function f(x) = x + 3 and g(x) = 3x2 – 1 are equal.
Let X = {–1, 0, 2, 5} and f : X → R Z: f(x) = x3 + 1. Then, write f as a set of ordered pairs.
Let A = {–2, –1, 0, 2} and f : A → Z: f(x) = x2 – 2x – 3. Find f(A).
Let f : R → R : f(x) = x2.
Determine (i) range (f) (ii) {x : f(x) = 4}
Let f : R → R : f(x) = x2 + 1. Find f–1 {10}.
Let f : R+ → R : f(x) = loge x. Find {x : f(x) = –2}.
Let A = {6, 10, 11, 15, 12} and let f : A → N : f(n) is the highest prime factor of n. Find range (f).
Find the range of the function f(x) = sin x.
Find the range of the function f(x) = |x|.
Write the domain and the range of the function,.
If then find dom (f) and range (f).
Let f = {(1, 6), (2, 5), (4, 3), (5, 2), (8, –1), (10, –3)} and g = {(2, 0), (3, 2), (5, 6), (7, 10), (8, 12), (10, 16)}.
Find (i) dom (f + g) (ii) dom .
If , find the value of .
If , where x ≠ –1 and f{f(x)} = x for x ≠ –1 then find the value of k.
Find the range of the function, .
Find the domain of the function, f(x) = log |x|.
If = for all x ϵ R – {0} then write an expression for f(x).
Write the domain and the range of the function, f(x) = .
Write the domain and the range of the function, f(x) = –|x|.
