Functions

Step 1: (A) has (2) elements and (B) gives (3) choices. Step 2: Each element has (3) possible images. Step 3: Total functions are (3^2=9).

Step 1: The number of functions is (n^m), where (m) is the number of domain elements and (n) is the number of codomain elements. Step 2: Here (m=2) and (n=4). Step 3: Therefore the number is (4^2=16).

Step 1: In a one-one function, the three inputs must have distinct images. Step 2: The first input has (3) choices, the second has (2), and the third has (1). Step 3: Total one-one functions are (3\cdot2\cdot1=6).

Step 1: The two inputs have different images, so the function is one-one. Step 2: Both elements (a) and (b) of (B) appear as images, so it is onto. Step 3: Being both one-one and onto, it is bijective.

Step 1: Both (1) and (2) map to (a), so the function is not one-one. Step 2: (c) is in the codomain but is not an image, so it is not onto. Step 3: Every input has one image, so it is a function, but neither one-one nor onto.

Step 1: Assume (f(a)=f(b)). Step 2: Then (3a+1=3b+1), so (3a=3b) and (a=b). Step 3: Hence different inputs have different images, so the function is one-one.

Step 1: In a one-one function, different inputs must have different images. Step 2: (2\ne -2), but (f(2)=0) and (f(-2)=0). Step 3: Therefore this function is not one-one.

Step 1: For onto, every real (y) must have a preimage. Step 2: From (x^3+1=y), we get (x=\sqrt[3]{y-1}). Step 3: This is real for every real (y), so the function is onto.

Step 1: (f(0)=|0-1|=1). Step 2: (f(2)=|2-1|=1). Step 3: Two different inputs have the same image, so the function is not one-one.

Step 1: An absolute value is always (0) or greater. Step 2: At (x=1), (|x-1|=0). Step 3: Therefore the range is ([0,\infty)).

Step 1: In an onto function, every element of the codomain becomes an image. Step 2: This means the range is equal to (B). Step 3: Therefore such a function is called onto.

Step 1: The main feature of a one-one function is that different inputs have different images. Step 2: This means two different elements do not map to the same image. Step 3: Therefore such a function is called one-one.

Step 1: (f(x)=x) is the identity function. Step 2: The identity function keeps every value unchanged. Step 3: Therefore its inverse is also (f^{-1}(x)=x).

Step 1: ((g\circ f)(x)=g(f(x))). Step 2: Put (f(x)=x+1) into (g). Step 3: (g(x+1)=(x+1)-1=x).

Step 1: (f(g(x))=f(\frac{x}{2})=x). Step 2: (g(f(x))=g(2x)=x). Step 3: Both composites give the identity function, so they are inverses.

Step 1: A bijective function has a well-defined inverse. Step 2: If the original function maps from (A) to (B), the inverse reverses the direction. Step 3: Therefore (f^{-1}) maps from (B) to (A).

Step 1: Put (x=2) in the function. Step 2: (f(2)=6\cdot2+2=12+2=14). Step 3: In a linear function, multiply first and then add.

Step 1: Put (x=2). Step 2: (f(2)=2^2+3\cdot2=4+6=10). Step 3: Do the square and multiplication in the correct order.

Step 1: Put (x=-2). Step 2: (f(-2)=2(-2)^2-1=2\cdot4-1=7). Step 3: The square of a negative number is positive.

Step 1: In an ordered pair, the first component is the input and the second is the image. Step 2: Since ((2,6)) is given, the image of (2) is (6). Step 3: Do not reverse the order of components.