One-one function

Step 1: The line (y=4) represents one output value. Step 2: Two intersections mean two different (x)-values give the same (y)-value. Step 3: By the horizontal line test, such a function is not one-one.

Step 1: Each horizontal line represents a fixed output value. Step 2: At most one intersection means that output comes from only one input. Step 3: This is the graphical test for one-one property.

Step 1: Assume ((g\circ f)(a)=(g\circ f)(b)). Step 2: From (g(f(a))=g(f(b))), first get (f(a)=f(b)), and then (a=b). Step 3: The composition of two one-one functions is one-one.

Step 1: (2^x) is an increasing function. Step 2: Adding (1) keeps different input values different. Step 3: Adding a constant to an increasing exponential function does not change one-one property.

Step 1: \(\left(\frac{1}{2}\right)^x\) is a decreasing exponential function. Step 2: A decreasing function does not give the same value for two different inputs. Step 3: An exponential function with base between (0) and (1) is also one-one.

Step 1: (\log_3 x) is increasing on the positive domain. Step 2: If (\log_3 a=\log_3 b), then (a=b). Step 3: A logarithmic function is one-one on its proper domain.

Step 1: (x^5) is an increasing odd-power function. Step 2: If (a^5=b^5), then (a=b). Step 3: Simple odd-power functions are one-one on real numbers.

Step 1: (1) and (-1) are different real numbers. Step 2: (1^6=(-1)^6=1), so the same value is obtained. Step 3: Even-power functions are not one-one on the whole real domain.

Step 1: On (x\geq0), (x^6) increases. Step 2: In this domain, different (x) values give different (x^6) values. Step 3: An even-power function can be one-one on the non-negative half-domain.

Step 1: The cube-root function increases on real numbers. Step 2: (x-2) also gives different values for different (x). Step 3: Such a translation of an increasing function preserves one-one property.

Step 1: (\sqrt{x-4}) increases on its domain. Step 2: If (\sqrt{a-4}=\sqrt{b-4}), then (a-4=b-4), so (a=b). Step 3: A square-root function is one-one on its proper domain.

Step 1: (f(0)=|-1|=1). Step 2: (f(1)=|1|=1), while (0\neq1). Step 3: In modulus functions, inner expressions with opposite signs can give the same output.

In a one-one function, two different elements of the domain cannot have the same image. Hence, if \(f(x_1)=f(x_2)\), it must follow that \(x_1=x_2\); this is exactly option A. Option B is the condition for an onto function, which is not required for a function to be one-one. Option C directly shows that the function is not one-one. Exam tip: If equal outputs force the inputs to be equal, the function is one-one.

Step 1: (f(0)=1). Step 2: (f(-1)=1-1+1=1), while (0\neq-1). Step 3: A quadratic function can give equal values on both sides of its axis.

In a one-one function, distinct inputs must have distinct images. Therefore, if \(f(a)=f(b)\), it must follow that \(a=b\); this is the defining condition for a one-one function. Option B is the condition for an onto function, which is different from being one-one. Exam tip: To test one-one-ness, check whether equal outputs force the inputs to be equal.

Step 1: From (f(a)=f(b)), we get (5-2a=5-2b). Step 2: This gives (-2a=-2b), hence (a=b). Step 3: A linear function with a negative coefficient can also be one-one.

Step 1: A constant function gives (7) for every input. Step 2: (f(1)=7) and (f(2)=7), while (1\neq2). Step 3: To disprove one-one, show equal outputs for different inputs.

Step 1: A one-one function needs (5) distinct images for the (5) elements of (A). Step 2: Therefore (B) must have at least (5) elements. Step 3: For finite sets, the codomain cannot be smaller than the domain for a one-one function.

Step 1: From (f(a)=f(b)), we get (\frac{a+2}{3}=\frac{b+2}{3}). Step 2: This gives (a+2=b+2), hence (a=b). Step 3: A linear rule with non-zero coefficient is one-one on rational numbers.

Step 1: Both (1) and (-1) are rational numbers. Step 2: (f(1)=1) and (f(-1)=1), while (1\neq-1). Step 3: The square function is not one-one on the whole rational domain.

Step 1: In the given domain, all (x\leq0). Step 2: On this domain, as (x) increases, (x^2) decreases, so different (x) values give different outputs. Step 3: The square function is also one-one on the non-positive half-domain.

Step 1: (f(1)=1+1=2). Step 2: (f(-1)=1+1=2), while (1\neq-1). Step 3: A function with only even powers can give the same value for (x) and (-x).

Step 1: In the given domain, (x\leq1), so (x-1\leq0). Step 2: On this part, as (x) increases, ((x-1)^2) decreases and different (x) values give different outputs. Step 3: A quadratic function can be one-one when restricted to one side of its axis.

Step 1: To show that a function is not one-one, find two different inputs with the same output. Step 2: \(f(2)=\frac{2}{5}\) and \(f\left(\frac{1}{2}\right)=\frac{\frac{1}{2}}{1+\frac{1}{4}}=\frac{2}{5}\), while \(2\neq\frac{1}{2}\). Step 3: In fractional functions, reciprocal pairs can often give the same value.

Step 1: The value of (x^3+x+2) increases continuously as (x) increases. Step 2: If (a>b), then (a^3-b^3+(a-b)>0), so (f(a)>f(b)). Step 3: A strictly increasing function is one-one.