Functions

Step 1: An invertible function must be one-one and onto. Step 2: If (a\neq 0), the linear function takes every real value exactly once. Step 3: If (a=0), it becomes constant and is not invertible.

Step 1: (f(2)=f(-2)), so the function is not one-one. Step 2: No negative real number can be its output, so it is not onto. Step 3: For absolute value functions, check both sign and range.

Step 1: (f^{-1}(8)) means the (x) for which (x^3=8). Step 2: Taking (x=2) gives (2^3=8). Step 3: In inverse values, we find the input from the output.

Step 1: ((f\circ g)(x)=f(g(x))). Step 2: Substitute (g(x)=x^2-1) into (f) to get (2(x^2-1)+1). Step 3: Simplifying gives (2x^2-1).

Step 1: Different values of (n) give different values of (n+1), so it is one-one. Step 2: The number (1) has no preimage in natural numbers. Step 3: For shift functions on natural numbers, the first value is often missed.

Step 1: (x+2) sends different integers to different integers. Step 2: For any (y\in\mathbb{Z}), choose (x=y-2\in\mathbb{Z}). Step 3: Shifts on integers are usually bijective.

Step 1: Since (x^2\geq 0), (1+x^2\geq 1). Step 2: The maximum value is (1) at (x=0), and the function approaches (0) but never reaches it. Step 3: For rational ranges, watch endpoint inclusion carefully.

Step 1: Put (y=\frac{1}{x}). Step 2: This gives (x=\frac{1}{y}). Step 3: Hence the inverse function is the same as the original function.

Step 1: Write (x^2-4x+7=(x-2)^2+3). Step 2: Since ((x-2)^2\geq 0), the minimum value is (3). Step 3: Completing the square quickly gives the minimum of a quadratic function.

Step 1: For an onto function, every element of (B) must be hit by some element of (A). Step 2: (A) has (3) elements but (B) has (4), so this is impossible. Step 3: For finite sets, the domain must not be smaller than the codomain for onto functions.

Step 1: The linear function (kx+7) is one-one when (k\neq 0). Step 2: To fail one-one, its slope must be (0). Step 3: At (k=0), the function becomes constant.

Step 1: (x^5+x) is strictly increasing on the real line, so it is one-one. Step 2: As (x\to\infty), the value goes to (\infty), and as (x\to-\infty), it goes to (-\infty). Step 3: An increasing function covering all real values is bijective.

Step 1: (f(1)=f(-1)), so it is not one-one. Step 2: Negative real numbers are not obtained as (x^4), so it is not onto. Step 3: For even powers, check both symmetry and range.

Step 1: From (y=3x-2), we get (x=\frac{y+2}{3}). Step 2: This matches the form of (g(y)), so (g) is the inverse function. Step 3: You can also verify an inverse by checking (f(g(x))=x) or (g(f(x))=x).

Step 1: The first element has (4) choices, the second has (3), and the third has (2). Step 2: Total one-one functions are (4\cdot3\cdot2=24). Step 3: For one-one functions, choose images without repetition.

Step 1: (\sin 0=\sin 2\pi), so the function is not one-one. Step 2: The range of (\sin x) is ([-1,1]), so it does not cover all of (\mathbb{R}). Step 3: For trigonometric functions, check periodicity and range.

Step 1: On ([0,\infty)), (x^2) is increasing, so it is one-one. Step 2: For every (y\geq0), (x=\sqrt{y}) lies in the domain. Step 3: Restricting the domain can make (x^2) bijective.

Step 1: Write (x^2-2x=(x-1)^2-1). Step 2: Since ((x-1)^2\geq0), the least value is (-1). Step 3: An upward-opening quadratic has range starting from its minimum value.

Step 1: The denominator of a fraction must not be zero. Step 2: (x+1=0) gives (x=-1). Step 3: To find the domain, first remove values that make the denominator zero.

Step 1: For every (x), the function value is (5). Step 2: Different inputs give the same output, so it is not one-one. Step 3: A constant function gives only one value, so it is usually not onto either.