One-one function

Step 1: In a one-one function, different inputs cannot give the same output. Step 2: If this happens, one-one property fails. Step 3: To prove not one-one, one counterexample is enough.

Step 1: In (x^2), (x) and (-x) give the same output. Step 2: For example, (f(3)=9) and (f(-3)=9). Step 3: The other functions are linear with non-zero coefficient of (x), so they are one-one.

Step 1: Here (2) and (3) are two different inputs. Step 2: Both have output (q), so an output is repeated. Step 3: A repeated output immediately breaks one-one property.

Step 1: The identity function sends every element to itself. Step 2: If (I_A(a)=I_A(b)), then (a=b) directly. Step 3: The identity function is the simplest and most important example of a one-one function.

Step 1: (f(4)=14) and (f(5)=15). Step 2: Different inputs are giving different outputs. Step 3: One example alone does not prove one-one, but the linear form (x+10) is fully one-one.

Step 1: (f(0)=|-1|=1). Step 2: (f(2)=|1|=1), while (0\neq2). Step 3: In absolute value functions, inputs at equal distance from the center can give the same output.

Step 1: A one-one function needs at least as many different codomain elements as domain elements. Step 2: Here (7\ge5), so assigning different outputs may be possible. Step 3: In counting questions, first count elements of domain and codomain.

Step 1: The domain has (6) elements, but the codomain has only (4). Step 2: It is impossible to send (6) different inputs to (4) different outputs without repetition. Step 3: If the domain is larger, one-one is impossible for finite sets.

Step 1: Assume (f(a)=f(b)). Step 2: From (\frac{a+1}{2}=\frac{b+1}{2}), we get (a+1=b+1), so (a=b). Step 3: A fractional-looking linear function can also be one-one.

Step 1: Because of the square, (1) and (-1) have the same square. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq-1). Step 3: Multiplying a square function by a positive constant still does not make it one-one on all (\mathbb{R}).

Step 1: One-one property does not allow the same output for different inputs. Step 2: Once (a\neq b) and (f(a)=f(b)) occur, the function is not one-one. Step 3: One violating pair rejects the whole one-one property.

Step 1: (f(2)=-1) and (f(4)=1). Step 2: These are different outputs, matching linear one-one behavior. Step 3: For a full proof, assume (f(a)=f(b)) and show (a=b).

Step 1: To prove one-one, assume (f(a)=f(b)). Step 2: From (3a+1=3b+1), we get (3a=3b), so (a=b). Step 3: For (mx+c) with (m\neq0), this method works quickly.

Step 1: The given domain contains only (0,1,2). Step 2: Their outputs are (0,1,4), with no repetition. Step 3: (x^2) is not one-one on every domain, so always check the given domain.

Step 1: This domain contains both (1) and (-1). Step 2: (f(1)=1) and (f(-1)=1), but (1\neq-1). Step 3: If the domain contains opposite numbers, (x^2) will not be one-one.

Step 1: In a one-one function, two different inputs cannot map to the same output. Step 2: So at most one element of (A) can map to one element of (B). Step 3: In arrow diagrams, count incoming arrows at each output to test one-one.

Step 1: Two arrows arriving at the same element means two different inputs have the same output. Step 2: This goes against the definition of one-one. Step 3: In diagram-based questions, decide quickly by checking incoming arrows at outputs.

Step 1: The cube-root function gives different outputs for different real inputs. Step 2: If (\sqrt[3]{a}=\sqrt[3]{b}), cubing both sides gives (a=b). Step 3: Keep the difference between cube root and square root in mind.

Step 1: (x^3) is one-one on all (\mathbb{R}). Step 2: Adding (2) shifts all outputs equally, but it does not make two different inputs equal. Step 3: Adding a constant to a cube function keeps it one-one.

Step 1: (f(0)=(1)^2=1). Step 2: (f(-2)=(-1)^2=1), while (0\neq-2). Step 3: A counterexample should clearly show two different inputs with the same output.

Step 1: Assume (f(a)=f(b)). Step 2: From (9-2a=9-2b), we get (-2a=-2b), hence (a=b). Step 3: Treat decreasing linear functions as one-one when the coefficient of (x) is non-zero.

Step 1: The outputs of the four inputs are (8,7,6,5). Step 2: No output is repeated. Step 3: Reverse order does not matter; if outputs are different, the function is one-one.

Step 1: (a) and (b) are different elements. Step 2: Both have output (2), so the same output is repeated. Step 3: Such repetition is not allowed in one-one property.

Step 1: This is a linear function of the form (mx+c). Step 2: Here (m=\frac{1}{3}\neq0), so different inputs give different outputs. Step 3: Do not be confused by a fractional coefficient; if it is non-zero, the linear function is one-one.

Step 1: To correct the mistake, we need an example where two different inputs give the same output. Step 2: (f(2)=4) and (f(-2)=4), but (2\neq-2). Step 3: One clear counterexample disproves the whole claim.