One-one function

Step 1: To test one-one, assume (f(a)=f(b)). Step 2: Then (2a+5=2b+5), so (a=b). Step 3: In exams, a linear function (mx+c) with (m\neq0) is one-one.

Step 1: A function is not one-one if two different inputs give the same output. Step 2: Here (1\neq-1), but (f(1)=1) and (f(-1)=1). Step 3: For square functions on all real numbers, always check opposite inputs.

In option A, the images of 1, 2 and 3 are a, b and c, so no two inputs have the same output. Hence it is one-one. Exam tip: check for repeated outputs; a repeat rejects one-one.

Step 1: A constant function gives the same output for every input. Step 2: (f(1)=7) and (f(2)=7), while (1\neq2). Step 3: In exams, a constant function is not one-one unless its domain has only one element.

Step 1: The outputs are (2,3,1). Step 2: No output is repeated, so different inputs have different outputs. Step 3: For finite sets, check repetition in the output list.

Step 1: For one-one, different inputs must have different outputs. Step 2: (1\neq2), but (f(1)=f(2)=1). Step 3: If two arrows go to the same output, the function is not one-one.

Step 1: (f(a)) means (a+1) and (f(b)) means (b+1). Step 2: So (f(a)=f(b)) becomes (a+1=b+1). Step 3: While proving one-one, first substitute correctly.

Step 1: (f(x)=3x-4) is linear with coefficient (3\neq0). Step 2: The other options can give the same output for two different inputs, such as (x) and (-x). Step 3: In easy questions, first identify linear and constant functions.

Step 1: Absolute value can give the same positive value for positive and negative inputs. Step 2: (2\neq-2), but (f(2)=2) and (f(-2)=2). Step 3: For ( |x| ) functions, quickly check opposite signs.

Step 1: Assume (f(a)=f(b)). Step 2: From (a+2=b+2), we get (a=b). Step 3: On natural numbers also, a function of the form (x+c) is one-one.

Step 1: If (f(a)=f(b)), then (2a=2b). Step 2: Dividing both sides by (2), we get (a=b). Step 3: Multiplication by a non-zero number gives different outputs for different inputs.

Step 1: (0x+4) equals (4) for every (x). Step 2: Different inputs, such as (1) and (3), give the same output (4). Step 3: In a linear form, coefficient (0) makes the function constant.

Step 1: One-one means no two different inputs give the same output. Step 2: This is exactly stated as different inputs having different outputs. Step 3: Remember it as no two arrows should land on the same output.

Step 1: The outputs for the three inputs are (1,3,5). Step 2: All are different, so there is no repetition. Step 3: For finite sets, the easiest one-one test is to check repeated outputs.

Step 1: The domain has (4) elements and the codomain has only (3) elements. Step 2: A one-one function needs a different output for every input, which is not enough here. Step 3: For finite sets, if the domain has more elements, a one-one function is not possible.

Step 1: The two domain elements have outputs (3) and (4). Step 2: These outputs are different, so the function is one-one. Step 3: Every codomain element need not be used; for one-one, only no repetition is needed.

Step 1: (x^3) has increasing behavior on real numbers. Step 2: If (a^3=b^3), then (a=b). Step 3: Remember the difference between (x^3) and (x^2); the cube function is one-one on (\mathbb{R}).

Step 1: Assume (f(a)=f(b)). Step 2: From (1-a=1-b), we get (a=b). Step 3: A decreasing linear function can also be one-one because different inputs give different outputs.

Step 1: In (x^2), opposite numbers give the same square. Step 2: (f(2)=5) and (f(-2)=5), while (2\neq-2). Step 3: Adding a constant does not make the square function one-one on all (\mathbb{R}).

Step 1: In positive real numbers, the opposite negative partner is not included. Step 2: If (a,b>0) and (a^2=b^2), then (a=b). Step 3: Changing the domain can change one-one property, so read the domain carefully.

Step 1: In a one-one function, different inputs have different outputs. Step 2: (1,2,3) are three different inputs, so (f(1)), (f(2)), (f(3)) will be different. Step 3: In such questions, directly applying the definition is the safest method.

Step 1: One-one means different inputs cannot give the same output. Step 2: Therefore, if (a\neq b), then (f(a)\neq f(b)). Step 3: Remember both forms: (f(a)=f(b)\Rightarrow a=b) and (a\neq b\Rightarrow f(a)\neq f(b)).

Step 1: This is a linear function and the coefficient of (x) is (-3). Step 2: Since (-3\neq0), (f(a)=f(b)) gives (a=b). Step 3: Even with a negative coefficient, a function can be one-one.

Step 1: (x^2-4x+4=(x-2)^2). Step 2: (f(1)=1) and (f(3)=1), while (1\neq3). Step 3: In perfect-square quadratic functions, equal values may occur on both sides of the vertex.

Step 1: Writing (f(a)=f(b)) gives (5a=5b). Step 2: Dividing both sides by (5), we get (a=b). Step 3: This result shows that (f(x)=5x) is one-one.