One-one function

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

Step 1: A constant function gives the same value for every input. Step 2: (f(1)=5) and (f(2)=5), but (1\neq 2). Step 3: If two different inputs have the same image, the function is not one-one.

Step 1: Check the images of all three inputs. Step 2: The images are (2), (3), and (1), with no repetition. Step 3: For finite sets, check repetition in images to test one-one.

Step 1: For one-one, every distinct input must have a distinct image. Step 2: Here (g(b)=2) and (g(c)=2), while (b\neq c). Step 3: A repeated image immediately disproves one-one.

Step 1: Assume (f(a)=f(b)). Step 2: Then (a^3=b^3), which gives (a=b) for real numbers. Step 3: The cube function is increasing on (\mathbb{R}), so it is one-one.

Step 1: The modulus function gives the same value to opposite inputs with equal distance from zero. Step 2: (f(3)=3) and (f(-3)=3), while (3\neq -3). Step 3: Finding two distinct inputs with the same image is the fastest test.

Step 1: In this domain, only (x\geq 0) is allowed. Step 2: If (a^2=b^2) and (a,b\geq 0), then (a=b). Step 3: Changing the domain can make the same formula one-one.

Step 1: In square-based functions, (x) and (-x) often give the same image. Step 2: (f(1)=3) and (f(-1)=3), while (1\neq -1). Step 3: For even powers on (\mathbb{R}), always test such pairs.

Step 1: Assume (f(a)=f(b)). Step 2: Then (7-4a=7-4b), so (-4a=-4b) and (a=b). Step 3: A linear function with non-zero slope is one-one.

Step 1: (x^2-6x+9=(x-3)^2). Step 2: (f(2)=1) and (f(4)=1), but (2\neq 4). Step 3: When a perfect square appears, test inputs equally distant from its center.

Step 1: Assume (f(a)=f(b)). Step 2: From (a^3+2=b^3+2), we get (a^3=b^3), so (a=b). Step 3: Adding a constant to a one-one function does not change one-one nature.

Step 1: A periodic function repeats values. Step 2: (\sin 0=0) and (\sin \pi=0), while (0\neq \pi). Step 3: For trigonometric functions, remember periodicity while checking one-one nature.

Step 1: Here the domain contains only (x\leq 0). Step 2: If (a^2=b^2) and (a,b\leq 0), then (a=b). Step 3: The square function becomes one-one on a half-domain.

Step 1: A modulus function gives equal values to inputs equally distant from its center. Step 2: (f(1)=1) and (f(3)=1), while (1\neq 3). Step 3: In (|x-a|), check equal distances on both sides of (a).

Step 1: Assume (f(m)=f(n)). Step 2: From (2m+1=2n+1), we get (m=n). Step 3: On integers also, a linear function with non-zero coefficient is one-one.

Step 1: Assume (f(m)=f(n)). Step 2: (m+4=n+4), so (m=n). Step 3: Adding the same number to different natural numbers preserves their difference.

Step 1: Natural numbers move in the positive direction. Step 2: If (m^2=n^2) and (m,n\in\mathbb{N}), then (m=n). Step 3: The square function is not one-one on (\mathbb{R}), but it is one-one on (\mathbb{N}).

Step 1: Check the images of all inputs. Step 2: (f(1)=1) and (f(2)=1), but (1\neq 2). Step 3: In a finite set, the same image for two inputs destroys one-one nature.

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: Even in fractional form, a linear function with non-zero coefficient is one-one.

Step 1: In an even-power function, (x) and (-x) give equal values. Step 2: (f(1)=1) and (f(-1)=1), while (1\neq -1). Step 3: On all real numbers, test opposite inputs for even powers.

Step 1: Assume (f(a)=f(b)), where (a,b>0). Step 2: From (\frac{1}{a}=\frac{1}{b}), we get (a=b). Step 3: In the reciprocal function, equal images occur only for equal inputs.

Step 1: Because of (x^2), opposite inputs can give the same value. Step 2: (f(1)=1) and (f(-1)=1), while (1\neq -1). Step 3: Whenever the denominator has (x^2), test positive and negative input pairs.

Step 1: Assume (f(a)=f(b)). Step 2: From (2a^3-5=2b^3-5), we get (a^3=b^3), so (a=b). Step 3: Multiplying a cubic function and adding a constant preserves one-one nature.

Step 1: (\cos x) is a periodic function. Step 2: (\cos 0=1) and (\cos 2\pi=1), while (0\neq 2\pi). Step 3: Periodic functions on all real numbers are generally not one-one.

Step 1: On the interval ([0,\pi]), (\cos x) is strictly decreasing. Step 2: A strictly decreasing function gives different values for different inputs. Step 3: Restricting the domain of a trigonometric function can make it one-one.