Functions

Step 1: Onto means every codomain element is used. Step 2: For finite equal sets, if two domain elements had the same image, some codomain element would be left unused. Step 3: Therefore an onto function from (A) to (A) is also one-one.

Step 1: ((f\circ f)(x)=f(f(x))). Step 2: (f(f(x))=\frac{1}{\frac{1}{x+1}+1}=\frac{x+1}{x+2}). Step 3: In fractional compositions, simplify the denominator carefully.

Step 1: Write (y=\sqrt{x-2}). Step 2: Squaring both sides gives (y^2=x-2), so (x=y^2+2). Step 3: When writing the inverse, the domain and range are interchanged.

Step 1: (f(0)=0), (f(\sqrt{3})=0), and (f(-\sqrt{3})=0), so the function is not one-one. Step 2: A cubic polynomial goes from very negative to very positive values, so every real value is attained. Step 3: A cubic function need not always be one-one.

Step 1: (A) has (5) elements and (B) has (3), so it is impossible to assign distinct images to all (5) elements. Step 2: But it is possible to cover all (3) elements of (B) using (5) elements of (A). Step 3: For finite sets, first compare cardinalities.

Step 1: (|x-2|\geq0) always. Step 2: Therefore (|x-2|+3\geq3), and at (x=2), the value is (3). Step 3: For modulus functions, the minimum is found by making the inside expression zero.

Step 1: If (x\neq0), then (\frac{1}{x}\neq0), so the function is well-defined. Step 2: (f(f(x))=\frac{1}{\frac{1}{x}}=x). Step 3: When (f\circ f) is the identity, the function is its own inverse.

Step 1: The function (x^5+x^3+x) is strictly increasing because its derivative (5x^4+3x^2+1) is always positive. Step 2: For very large negative (x), the value becomes very negative, and for very large positive (x), the value becomes very positive, so every real value is attained. Step 3: For increasing odd-power polynomials, check both one-one and onto carefully.