One-one function

Step 1: Assuming (f(a)=f(b)) gives (5a=5b). Step 2: Dividing both sides by (5) gives (a=b). Step 3: Multiplication by a non-zero number preserves one-one property.

Step 1: (f(2)=|2-3|=1). Step 2: (f(4)=|4-3|=1), although (2\neq4). Step 3: In modulus functions, points equally far from the center can give the same value.

Step 1: For (x\geq3), (|x-3|=x-3). Step 2: (x-3) is an increasing linear function. Step 3: Restricting the domain can make a modulus function one-one.

Step 1: (f(0)=0). Step 2: (f(2)=4-4=0), while (0\neq2). Step 3: In a quadratic function, equal values can occur on both sides of the axis.

Step 1: (x^2-2x=(x-1)^2-1). Step 2: For (x\geq1), ((x-1)^2) increases, so different (x) values give different outputs. Step 3: A quadratic function can be one-one on one side of its axis.

Step 1: One-one property is about comparing inputs and outputs. Step 2: If (f(a)=f(b)) leads to (a=b), the function is one-one. Step 3: Finding only one function value is not a proof of one-one.

Step 1: (x^3) is an increasing function. Step 2: Adding (2) keeps different input values different. Step 3: Adding a constant to a one-one function does not change one-one property.

Step 1: (x^3) gives different values for different real inputs. Step 2: Multiplying by (2) and subtracting (1) does not change the equality pattern. Step 3: Linear transformations of a basic cubic function remain one-one.

Step 1: (x^2+4x+4=(x+2)^2). Step 2: (f(-1)=1) and (f(-3)=1), although (-1\neq-3). Step 3: The perfect square form helps find two inputs with the same output.

Step 1: (\sin x) is a periodic function. Step 2: (\sin 0=0) and (\sin \pi=0), while (0\neq\pi). Step 3: Periodic functions are usually not one-one on the whole real domain.

Step 1: On the given interval, (\sin x) is increasing. Step 2: In this interval, two different (x) values do not give the same sine value. Step 3: Trigonometric functions can be one-one on suitable restricted intervals.

Step 1: (\cos x) is periodic. Step 2: (\cos 0=1) and (\cos 2\pi=1), while (0\neq2\pi). Step 3: If different angles give the same value, the function is not one-one.

Step 1: On ([0,\pi]), (\cos x) is decreasing. Step 2: A decreasing function gives different outputs for different inputs. Step 3: Being strictly decreasing is also a strong sign of one-one property.

Step 1: In a one-one function, two different elements cannot have the same image. Step 2: Therefore selected images are not repeated. Step 3: Counting one-one functions uses the idea of permutations.

Step 1: The first element has (4) choices. Step 2: The second element has (3) remaining choices because repetition is not allowed. Step 3: Total one-one functions are (4\cdot3=12).

Step 1: The first element has (3) choices. Step 2: Then (2) choices and then (1) choice remain. Step 3: The total number is (3\cdot2\cdot1=6).

Step 1: A one-one function needs (4) distinct images. Step 2: The codomain has only (2) elements, so this is impossible. Step 3: If the domain is larger than the codomain, the number of one-one functions may be zero.

Step 1: In a one-one function, each domain element has a distinct image. Step 2: Therefore the range has as many elements as the domain. Step 3: This fact is very useful in counting problems on finite sets.

Step 1: Assume (f(a)=f(b)). Step 2: (\frac{a+1}{2}=\frac{b+1}{2}) gives (a+1=b+1), so (a=b). Step 3: A fixed denominator does not break one-one property.

Step 1: From (f(a)=f(b)), we get (\frac{1}{a}=\frac{1}{b}). Step 2: Since (a\neq0) and (b\neq0), this gives (a=b). Step 3: The reciprocal function is one-one on its proper domain.

Step 1: (x^2+2x+5=(x+1)^2+4). Step 2: (f(0)=5) and (f(-2)=5), while (0\neq-2). Step 3: A quadratic function is generally not one-one on the whole real domain.

Step 1: (f(x)=(x+1)^2+4). Step 2: For (x\geq-1), ((x+1)^2) increases. Step 3: On one side of its axis, a quadratic function can be one-one.

Step 1: (f(x)=x+1) is linear and the coefficient of (x) is (1\neq0). Step 2: The other options can give equal outputs for different inputs or are constant. Step 3: In easy questions, identify linear and constant functions first.

Step 1: Assuming (f(a)=f(b)), we get (6-2a=6-2b). Step 2: Simplifying gives (-2a=-2b), hence (a=b). Step 3: A linear function is one-one when the coefficient of (x) is not zero.

Step 1: Each element of the domain has exactly one image, so it is a function. Step 2: But both (2) and (4) have image (7). Step 3: In a finite set, repeated images mean the function is not one-one.