Question

Example: For A = {1, 2, 3, 4, 5) and B = {w, x, y, z), let f: A → B be given by f = {(1, w) (2, x), (3, x), (4, y), (5, y)). Then for A₁ = (1), A₂ = {1, 2), A3 = (1, 2, 3), A4 = (2, 3) and As = {2, 3, 4, 5), we find the following corresponding images under f:

ƒ(A1)=
ƒ(A2)=
ƒ (A3)=
ƒ(A4)=
ƒ(A5)=
Example: Let g: R-->R, where g(x)=x², VER. Find the range of g. Find g(Z). ​

Answer 1

The given function f maps elements from set A to B.

The images of the sets A₁, A₂, A₃, A₄, and As under f are determined.

The range of the function g(x) = x² is found to be [0, ∞), and g(Z) includes the squares of integers.

Step-by-step explanation:

The given function is f: A → B, where A = {1, 2, 3, 4, 5} and B = {w, x, y, z}.

The function is defined as f = {(1, w), (2, x), (3, x), (4, y), (5, y)}.

To find the images of the sets A₁, A₂, A₃, A₄, and As under f:

ƒ(A₁) = {w} (image of 1 under f)

ƒ(A₂) = {x} (image of 1 and 2 under f)

ƒ(A₃) = {x} (image of 1, 2, and 3 under f)

ƒ(A₄) = {x, y} (image of 2 and 3 under f)

ƒ(As) = {x, y} (image of 2, 3, 4, and 5 under f)

The range of the function g(x) = x² is all real numbers greater than or equal to 0. Therefore, the range of g is [0, ∞).

The set Z refers to the set of all integers. To find g(Z), we substitute each integer value into the function g(x) = x²:

g(Z) = {0, 1, 4, 9, 16, ...} (squares of all integers).