One-one function

Step 1: Find (f'(x)=1-\cos x). Step 2: (1-\cos x\ge 0) for every real (x), and the function does not turn back. Step 3: With a non-negative derivative, check whether equal values can repeat before deciding one-one nature.

Step 1: The three elements of (A) must be assigned distinct elements of (B). Step 2: The choices are (5\cdot4\cdot3=60). Step 3: The number of all functions would be (5^3), but one-one functions cannot repeat images.

Step 1: In (x^2), (x) and (-x) can give the same value. Step 2: On ([0,\infty)), the negative side is removed and the function is strictly increasing. Step 3: To make the square function one-one, keep the domain on one side only.

Step 1: The domain contains only zero and negative numbers. Step 2: On this domain, as (x) increases, (x^2) decreases, so the same value is not obtained at two different (x)-values. Step 3: A strictly decreasing function is also one-one.

Step 1: (2^x) is an exponential function with base greater than (1). Step 2: As (x) increases, (2^x) increases strictly. Step 3: A strictly increasing exponential function is one-one on its domain.

Step 1: A constant function gives the same value for every (x). Step 2: (f(1)=5) and (f(2)=5), while (1\ne2). Step 3: A constant function on a domain with more than one element is never one-one.

Step 1: The derivative is (f'(x)=3x^2-12x+12=3(x-2)^2). Step 2: It is zero or positive everywhere, and (f(x)=(x-2)^3+9), which is one-one. Step 3: Rewriting a cubic can sometimes give a clearer solution than derivative alone.

Step 1: The vertex of ((x-1)^2) is at (x=1). Step 2: The interval ([0,2]) contains points on both sides of the vertex, such as (0) and (2), both giving value (1). Step 3: If an interval contains points equally placed around the vertex, a quadratic is not one-one.

Step 1: For (x<0), (|x|=-x), so (f(x)=0). Step 2: All negative (x)-values give (0), for example (f(-1)=f(-2)=0). Step 3: Constant behaviour on a large part destroys one-one nature.

Step 1: The domain of (\sqrt{x}) is ([0,\infty)). Step 2: On this domain, as (x) increases, (\sqrt{x}) also increases. Step 3: The square-root function is one-one on its natural domain.

Step 1: Write (x^2+x=(x+\frac{1}{2})^2-\frac{1}{4}). Step 2: The vertex is (x=-\frac{1}{2}), and the function increases to the right of it. Step 3: Completing the square quickly gives the one-one interval.

Step 1: Suppose (f(a)=f(b)). Step 2: Then (g(f(a))=g(f(b))), so ((g\circ f)(a)=(g\circ f)(b)). Since (g\circ f) is one-one, (a=b). Step 3: In composition questions, apply the definition directly.

Step 1: Assume ((g\circ f)(a)=(g\circ f)(b)). Step 2: Then (g(f(a))=g(f(b))), and since (g) is one-one, (f(a)=f(b)). Since (f) is one-one, (a=b). Step 3: The composition of two one-one functions is one-one.

Step 1: Assume (f(a)=f(b)). Step 2: (\frac{a+1}{a-1}=\frac{b+1}{b-1}) gives ((a+1)(b-1)=(b+1)(a-1)), leading to (a=b). Step 3: For fractional linear functions, cross-multiplication is a good way to test one-one nature.

Step 1: (f'(x)=3x^2+2x+1). Step 2: Its discriminant is (2^2-4\cdot3\cdot1=-8<0) and the leading coefficient is positive, so (f'(x)>0) for all (x). Step 3: If the derivative is always positive, the function is one-one.

Step 1: The domain has (4) elements and the codomain has (3) elements. Step 2: A one-one function needs four distinct images, which is impossible with only (3) codomain elements. Step 3: If the codomain is smaller than the domain, no one-one function exists.

Step 1: Check values at two different inputs. Step 2: (f(0)=0) and (f(3)=27-27=0), while (0\ne3). Step 3: A cubic function is not always one-one; turning behaviour can create repeated values.

Step 1: For a higher-degree polynomial, studying monotonicity is useful. Step 2: Find (f'(x)) and check whether its sign changes. Step 3: In difficult one-one questions, the derivative method gives a systematic path.

Step 1: (\lfloor x\rfloor) gives the greatest integer less than or equal to the number. Step 2: (f(1.2)=1) and (f(1.8)=1), while (1.2\ne1.8). Step 3: Step-like functions often map many inputs to the same output.

Step 1: (\lceil x\rceil) gives the smallest integer greater than or equal to (x). Step 2: (\lceil1.1\rceil=2) and (\lceil1.9\rceil=2), while the inputs are different. Step 3: For integer-valued functions, nearby decimal values are useful for testing one-one nature.

Step 1: For (-1\le x\le1), (|x-1|+|x+1|=2). Step 2: Many different (x)-values in this interval give function value (2). Step 3: If a function is constant on an interval, it cannot be one-one.

Step 1: Write (x^3+3x^2+3x+5=(x+1)^3+4). Step 2: ((x+1)^3) is a cubic form that is strictly increasing on all real numbers. Step 3: Recognizing a shifted cube helps decide one-one nature quickly.

Step 1: (f(1)=3), (f(2)=2), (f(3)=1). Step 2: All three images are distinct, so no two different inputs have the same image. Step 3: For finite sets, a value table quickly checks one-one nature.

Step 1: In a square function, opposite numbers have the same square. Step 2: (f(2)=4-1=3) and (f(-2)=4-1=3), while (2\ne-2). Step 3: For quadratic functions, pairs like (x) and (-x) often give a quick counterexample.

Step 1: In a one-one function, every distinct element of (A) has a distinct image. Step 2: Therefore, (n) elements of (A) produce (n) distinct elements in (f(A)). Step 3: A one-one function preserves the size of the image set equal to the domain size.