Let f : R → R and g : C → C be two functions defined as f(x) = x2 and g(x) = x2. Are they equal functions?

Q9 of 118 Page 4

Let f : R → R and g : C → C be two functions defined as f(x) = x² and g(x) = x². Are they equal functions?

Given f : R → R ∋ f(x) = x² and g : R → R ∋ g(x) = x²

As f is defined from R to R, the domain of f = R.

As g is defined from C to C, the domain of g = C.

Two functions are equal only when the domain and codomain of both the functions are equal.

In this case, the domain of f ≠ domain of g.

Thus, f and g are not equal functions.

Let f: R⁺→ R, where R⁺ is the set of all positive real numbers, be such that f(x) = log_ex. 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}}

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)}