One-one function

Step 1: To test one-one, assume (f(a)=f(b)). Step 2: (3a-7=3b-7) gives (a=b). Step 3: In exams, a linear function with non-zero coefficient of (x) is one-one.

Step 1: To disprove one-one, it is enough to find two different inputs with the same output. Step 2: (f(1)=2) and (f(-1)=2), while (1\ne -1). Step 3: For even power functions, always check positive and negative inputs.

Step 1: Assume (f(a)=f(b)). Step 2: (a^3-4=b^3-4) gives (a^3=b^3), hence (a=b). Step 3: A strictly increasing function like (x^3) remains one-one on real numbers.

Step 1: Absolute value can give the same output for positive and negative inputs. Step 2: (f(2)=2) and (f(-2)=2), while (2\ne -2). Step 3: In one-one tests, search for two different inputs with equal outputs.

Step 1: The domain is only (x\ge 0). Step 2: On this domain, (x^2) is increasing, so (a^2=b^2) implies (a=b). Step 3: Changing the domain can change whether a function is one-one.

Step 1: A constant function gives the same output for every input. Step 2: (f(1)=5) and (f(2)=5), but (1\ne 2). Step 3: A constant function can be one-one only if the domain has a single element.

Step 1: Write (f(a)=f(b)). Step 2: (2a^3+1=2b^3+1) gives (a^3=b^3), so (a=b). Step 3: This is the safest method to prove one-one in written exams.

Step 1: Each input has one output, so it is a function. Step 2: Inputs (1) and (3) both have output (4). Step 3: If two different inputs share an output, the function is not one-one.

Step 1: Every input has exactly one output. Step 2: The outputs (b,c,a) are all distinct, so no two inputs have the same output. Step 3: For finite sets, checking repeated outputs quickly identifies one-one behavior.

Step 1: Assume (f(m)=f(n)). Step 2: (2m+3=2n+3) gives (m=n). Step 3: Even on integers, a linear function with non-zero coefficient is one-one.

Step 1: Integers include both positive and negative values. Step 2: (f(2)=4) and (f(-2)=4), while (2\ne -2). Step 3: Before calling (n^2) one-one on integers, test values with opposite signs.

Step 1: Assume (f(a)=f(b)). Step 2: (\frac{1}{a}=\frac{1}{b}) with (a,b>0) gives (a=b). Step 3: A reciprocal function gives distinct outputs for distinct inputs in its valid domain.

Step 1: The domain contains both (1) and (-1). Step 2: (f(1)=1) and (f(-1)=1), while (1\ne -1). Step 3: In reciprocal square functions, changing the sign can keep the output same.

Step 1: A quadratic function can give equal outputs for two different inputs due to symmetry. Step 2: (f(1)=4) and (f(3)=4), while (1\ne 3). Step 3: Completing the square or substituting two values can quickly disprove one-one.

Step 1: We can write (f(x)=(x-2)^2+3). Step 2: For (x\ge 2), ((x-2)^2) increases, so different inputs give different outputs. Step 3: Restricting a quadratic function to one side of its vertex can make it one-one.

Step 1: Assume (a>b). Step 2: Then (a^3>b^3) and (a>b), so (a^3+a>b^3+b). Step 3: A strictly increasing function is one-one.

Step 1: (f(x)=x^3-x=x(x-1)(x+1)). Step 2: (f(-1)=0), (f(0)=0), and (f(1)=0), while the inputs are different. Step 3: If many inputs give the same output, the function is not one-one.

Step 1: Assume (f(a)=f(b)). Step 2: (7-2a=7-2b) gives (a=b). Step 3: A decreasing linear function is also one-one because it gives distinct outputs for distinct inputs.

Step 1: Trigonometric functions are periodic. Step 2: (\sin 0=0) and (\sin \pi=0), while (0\ne \pi). Step 3: Periodic functions are generally not one-one on the entire real domain.

Step 1: On the given interval, (\sin x) is strictly increasing. Step 2: So different (x)-values give different (\sin x)-values. Step 3: A trigonometric function can become one-one when its domain is restricted properly.

Step 1: On ([0,\pi]), (\cos x) is strictly decreasing. Step 2: A strictly decreasing function does not give the same output for two different inputs. Step 3: Strictly increasing or decreasing behavior is a good sign of one-one nature.

Step 1: (\cos x) is periodic. Step 2: (\cos 0=1) and (\cos 2\pi=1), while (0\ne 2\pi). Step 3: Periodicity on the whole real domain strongly proves the function is not one-one.

Step 1: (e^x) is strictly increasing on real numbers. Step 2: If (e^a=e^b), then (a=b). Step 3: The increasing nature of an exponential function helps prove one-one behavior.

Step 1: (\log x) is increasing on its domain ((0,\infty)). Step 2: If (\log a=\log b), then (a=b). Step 3: A logarithmic function is one-one on its valid domain.

Step 1: Values equally distant from (x=2) give the same output. Step 2: (f(1)=1) and (f(3)=1), while (1\ne 3). Step 3: For absolute value functions, test points on both sides of the center.