One-one function

Step 1: To test one-one, assume (f(a)=f(b)). Step 2: From (3a-7=3b-7), we get (a=b), so different inputs have different images. Step 3: In exams, a linear function (mx+c) with (m\neq 0) is one-one.

Step 1: In a one-one function, different inputs must have different images. Step 2: Here (f(2)=4) and (f(-2)=4), while (2\neq -2). Step 3: If two different inputs give the same image, the function is not one-one.

Step 1: Let (f(a)=f(b)). Step 2: From (5-2a=5-2b), we get (a=b). Step 3: A negative slope is not a problem; a linear function is one-one if its slope is non-zero.

Step 1: (x^3) is an increasing function. Step 2: If (a^3=b^3), then (a=b). Step 3: On the whole real set, (x^3) is a good example of a one-one function.

Step 1: In an absolute value function, opposite signed inputs can have the same image. Step 2: (f(3)=3) and (f(-3)=3), but (3\neq -3). Step 3: Such an example quickly shows that the function is not one-one.

Step 1: Check by assuming (f(a)=f(b)). Step 2: From (a+4=b+4), we get (a=b). Step 3: Adding the same constant keeps different values different.

Step 1: For one-one nature, it is not necessary to get every value of the codomain. Step 2: If (f(a)=f(b)), then (2a=2b), so (a=b). Step 3: The range may be smaller, yet the function can still be one-one.

Step 1: In a function involving (x^2), (x) and (-x) may have the same image. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq -1). Step 3: One counterexample is enough to disprove one-one nature.

Step 1: Here the inputs are only (0) and positive numbers. Step 2: On this domain, (x^2) increases, so (a^2=b^2) gives (a=b). Step 3: One-one nature depends on the domain.

Step 1: A constant function gives the same value for every input. Step 2: (f(1)=7) and (f(2)=7), but (1\neq 2). Step 3: If the domain has more than one element, a constant function is not one-one.

Step 1: The images of the three inputs are (2,3,1). Step 2: No two different inputs give the same image. Step 3: For a finite set, one-one nature can be checked quickly by comparing images.

Step 1: (1) and (2) are different inputs. Step 2: Both have image (2), so different inputs give the same image. Step 3: In a finite table, a repeated image disproves one-one nature.

Step 1: (x^3) is increasing on real numbers. Step 2: Adding the constant (1) does not change one-one nature. Step 3: From (a^3+1=b^3+1), we get (a=b).

Step 1: In a squared function, points on both sides of the center may give the same image. Step 2: (f(1)=1) and (f(3)=1), while (1\neq 3). Step 3: Equal images for different inputs mean the function is not one-one.

Step 1: Let (f(a)=f(b)), where (a,b>0). Step 2: From (\frac{1}{a}=\frac{1}{b}), we get (a=b). Step 3: In this reciprocal form, equal images lead to equal inputs.

Step 1: Here both (x) and (-x) are in the domain. Step 2: (f(2)=\frac{1}{4}) and (f(-2)=\frac{1}{4}), while (2\neq -2). Step 3: Different inputs with the same image prove that the function is not one-one.

Step 1: Writing (f(a)=f(b)) gives (2a+9=2b+9). Step 2: Simplifying gives (2a=2b), so (a=b). Step 3: This is the algebraic test for a one-one function.

Step 1: In (x^2) and (|x|), (x) and (-x) can give the same image. Step 2: (f(x)=6) is constant. Step 3: (4x-1) has non-zero slope, so it is one-one.

Step 1: The odd power (x^5) keeps increasing on real numbers. Step 2: If (a^5=b^5), then (a=b). Step 3: Simple odd-power functions are often one-one on the whole real set.

Step 1: In an even power, (x) and (-x) have the same image. Step 2: (f(2)=16) and (f(-2)=16). Step 3: Even-power functions are generally not one-one on all of (\mathbb{R}).

Step 1: (x^3) is an increasing function. Step 2: Multiplication by (2) and subtracting a constant do not break the order. Step 3: From (2a^3-5=2b^3-5), we get (a=b).

Step 1: In a one-one function, different inputs cannot give the same image. Step 2: Its algebraic form is (f(a)=f(b)\Rightarrow a=b). Step 3: Remembering the definition helps solve many questions quickly.

Step 1: The images of all four inputs are different. Step 2: No image is repeated. Step 3: In a finite function, if images are not repeated, the function is one-one.

Step 1: In an absolute value function, inputs at equal distance from the center give the same image. Step 2: (f(0)=1) and (f(2)=1), while (0\neq 2). Step 3: Two different inputs with the same image prevent one-one nature.

Step 1: On ([0,\infty)), (x^2) increases continuously. Step 2: In this domain, two different non-negative numbers will not have the same square. Step 3: One-one nature can change with the domain, so read the domain carefully.