One-one function

Step 1: In a one-one function different inputs must give different images. Step 2: From (3x_1-5=3x_2-5), we get (x_1=x_2). Step 3: A linear function with non-zero coefficient of (x) is one-one.

Step 1: The domain contains only (0) and positive real numbers. Step 2: If (x_1^2=x_2^2) and (x_1,x_2\ge 0), then (x_1=x_2). Step 3: Changing the domain can make the same rule one-one.

Step 1: The cube function preserves order on real numbers. Step 2: From (2x_1^3+1=2x_2^3+1), we get (x_1^3=x_2^3), so (x_1=x_2). Step 3: Simple increasing odd-power functions are often one-one.

Step 1: In an absolute value function inputs equally distant from the center can give the same value. Step 2: (1\neq 3), but (f(1)=1) and (f(3)=1). Step 3: One repeated image is enough to reject one-one nature.

Step 1: In a one-one function two different elements of (A) cannot map to the same element of (B). Step 2: Here (1\neq 3), but both have image (a). Step 3: In mapping-table questions first look for repeated images.

Step 1: Do not decide only by seeing the cubic term. Step 2: (f(-1)=0), (f(0)=0), and (f(1)=0). Step 3: Same image for different inputs immediately disproves one-one nature.

Step 1: (2^x) is strictly increasing on real numbers. Step 2: If (2^{x_1}=2^{x_2}), then (x_1=x_2). Step 3: An exponential function with base greater than (1) is one-one.

Step 1: Trigonometric functions repeat values over the full real domain. Step 2: (0\neq \pi), but (\sin 0=0) and (\sin \pi=0). Step 3: Periodic functions are usually not one-one on the whole real domain.

Step 1: On the given interval (\sin x) is strictly increasing. Step 2: So two different (x)-values do not give the same sine value. Step 3: Restricting the domain is very important for trigonometric functions.

Step 1: Test the function by assuming equal images. Step 2: If (\frac{1}{x_1}=\frac{1}{x_2}), then (x_1=x_2). Step 3: For reciprocal functions always check the domain first.

Step 1: Zero is removed from the domain, so the function is well-defined. Step 2: From (\frac{1}{x_1}=\frac{1}{x_2}), we get (x_1=x_2). Step 3: Negative and positive parts together do not destroy injectivity here.

Step 1: Injectivity can be checked by finding repeated images. Step 2: \(f(2)=\frac{2}{5}\) and \(f\left(\frac{1}{2}\right)=\frac{2}{5}\), while \(2\neq \frac{1}{2}\). Step 3: Do not assume a rational expression is one-one without testing.

Step 1: Both (x^5) and (x) increase on real numbers. Step 2: Their sum also increases, so different inputs give different values. Step 3: A strictly increasing function is one-one.

Step 1: The greatest integer function maps many real numbers to the same integer. Step 2: For example (f(1.2)=1) and (f(1.8)=1), while (1.2\neq 1.8). Step 3: Step-like functions are generally not one-one.

Step 1: A one-one function needs distinct images in (B) for distinct elements of (A). Step 2: (A) has (4) elements but (B) has only (3) elements. Step 3: If the domain is larger than the codomain, a one-one function may be impossible.

Step 1: For a one-one function the three elements of (A) need three distinct images from (B). Step 2: The number is (^{5}P_{3}=5\cdot4\cdot3=60). Step 3: Use permutations when counting one-one functions.

Step 1: Both sets have (4) elements. Step 2: For a one-one function all (4) elements of (B) are arranged distinctly, so the number is (4!=24). Step 3: Between finite sets of equal size a one-one function is also onto.

Step 1: Assume ((g\circ f)(x_1)=(g\circ f)(x_2)). Step 2: Since (g) is one-one, (f(x_1)=f(x_2)), and since (f) is one-one, (x_1=x_2). Step 3: The composition of two one-one functions is one-one.

Step 1: If (g\circ f) is one-one, the first function (f) cannot send two different inputs to the same value. Step 2: If (f(x_1)=f(x_2)), then ((g\circ f)(x_1)=(g\circ f)(x_2)), so (x_1=x_2). Step 3: If (g\circ f) is one-one, then (f) must be one-one.

Step 1: (g\circ f) only uses the behavior of (g) on the range of (f). Step 2: (g) may fail to be one-one on its full domain but still behave distinctly on the range of (f). Step 3: In composition questions distinguish the whole domain from the used part.

Step 1: In a one-one function every image has only one original input. Step 2: So when reversed, each element of (f(A)) gets exactly one element of (A). Step 3: Injectivity is the key condition for an inverse to be a function.

Step 1: In a linear function (a) controls the change with (x). Step 2: If (a\neq0), then (ax_1+b=ax_2+b) gives (x_1=x_2). Step 3: If (a=0), the function becomes constant and is not one-one.

Step 1: (x^2+2x+3=(x+1)^2+2), so the graph is a parabola. Step 2: Inputs equally distant from (-1) give the same value. Step 3: A parabola is generally not one-one on the whole real domain.

Step 1: (x^2-2x=(x-1)^2-1). Step 2: On the domain (x\ge1), this function increases, so different inputs give different values. Step 3: A quadratic can become one-one when restricted to one side of its vertex.

Step 1: The vertex of (x^2-2x=(x-1)^2-1) is at (x=1). Step 2: On (x\le1), the function moves in one direction, so no value repeats. Step 3: A quadratic restricted to one side of its vertex is one-one.