Solution of Chapter 3. Functions (RS Aggarwal - Mathematics Book)

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) = log_ex. 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) =2^x. Find

(i) range (f)

(ii) {x : f(x) = 1}.

(iii) Find out whether f(x + y) = f(x). f(y) for all x, y ϵ R.

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

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.

(i)

(ii)

(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:

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:

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) =

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:

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) = 1 – |x – 2|

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:

Find the domain and the range of each of the following real function:

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) = x^1/3 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

(i) (f + g) (x)

(ii) (f – g) (x)

(iii) (fg) (x)

(iv) (f/g)(x)

. Let f: R → R: f(x) = x³ + 1 and g: R → R: g(x) = (x + 1). Find:

(i) (f + g) (x)

(ii) (f – g) (x)

(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)

(iii)

. Let f:(2, ∞) → R: f(x) = and g: (2, ∞) → R: g(x) = Find:

(i) (f + g) (x)

(ii) (f - g) (x)

(iii) (fg) (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) = log_e 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,.

Write the domain and the range of the function, f(x) = .

Write the domain and the range of the function, f(x) = –|x|.