One-one function

Step 1: For (x\geq0), (|x|=x). Step 2: Thus (f(x)=2x), which gives different values for different inputs. Step 3: Changing the domain can change the one-one nature of a modulus function.

Step 1: (f(0)=0). Step 2: (f(1)=1-1=0), while (0\neq1). Step 3: Not every cubic-looking function is one-one; check the rule.

Step 1: This function increases continuously as (x) increases. Step 2: If (a>b), then (a^3+4a) is greater than (b^3+4b). Step 3: A strictly increasing function is one-one.

Step 1: (x^3) increases, so (-x^3) decreases. Step 2: (5-x^3) is decreasing and gives different outputs for different inputs. Step 3: A strictly decreasing function is also one-one.

Step 1: From (f(a)=f(b)), we get (\frac{2}{a}=\frac{2}{b}). Step 2: Since (a,b>0), this gives (a=b). Step 3: A reciprocal-type rule is one-one on its proper domain.

Step 1: From (f(a)=f(b)), we get (\frac{1}{a-2}=\frac{1}{b-2}). Step 2: Since (a\neq2) and (b\neq2), we get (a-2=b-2), hence (a=b). Step 3: The value making the denominator zero must be removed from the domain.

Step 1: (\sin 0=0). Step 2: (\sin \pi=0), while (0\neq\pi). Step 3: In periodic functions, different angles can give the same value.

Step 1: (\cos x) is periodic. Step 2: (\cos 0=1) and (\cos 2\pi=1), while (0\neq2\pi). Step 3: Equal values at different inputs destroy one-one property.

Step 1: (2^x) is an increasing exponential function. Step 2: Different real (x) values give different values of (2^x). Step 3: An exponential function with base greater than (1) is one-one.

Step 1: \(\left(\frac{1}{3}\right)^x\) is a decreasing exponential function. Step 2: A decreasing function gives different values for different inputs. Step 3: An exponential function with base between (0) and (1) is also one-one.

Step 1: (\log_2 x) increases on the positive domain. Step 2: If (\log_2 a=\log_2 b), then (a=b). Step 3: A logarithmic function is one-one on its proper domain.

Step 1: A constant function sends all inputs to the same image. Step 2: Since (A) has more than one element, different inputs will have the same image. Step 3: Do not mistake a constant function for a one-one function.

Step 1: In a one-one function, images are not repeated. Step 2: For the three elements of (A), three distinct elements of (B) must be chosen. Step 3: In finite sets, distinct images are the key to one-one property.

Step 1: A horizontal line represents the same (y)-value. Step 2: Two intersections mean two different (x)-values have the same (y)-value. Step 3: The horizontal line test is useful for checking one-one property from a graph.

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

Step 1: (f(2)=3\cdot4=12). Step 2: (f(-2)=3\cdot4=12), while (2\neq-2). Step 3: Even with a coefficient, a square can give the same value for (x) and (-x).

Step 1: In the given domain, (x) is not negative. Step 2: For (x\geq0), (3x^2) increases. Step 3: Restricting a square function to the non-negative side makes it one-one.

Step 1: (f(2)=0). Step 2: (f(-2)=0), while (2\neq-2). Step 3: Square-based functions are not one-one on the whole real domain.

Step 1: Putting (f(a)=f(b)) gives (\frac{a}{2}+6=\frac{b}{2}+6). Step 2: This gives (\frac{a}{2}=\frac{b}{2}), hence (a=b). Step 3: A fractional coefficient still gives one-one property if it is non-zero.

Step 1: (0x-4=-4) is the same for every (x). Step 2: (f(1)=-4) and (f(2)=-4), while (1\neq2). Step 3: If the coefficient of (x) is zero, the function becomes constant.

Step 1: Each element of the domain has exactly one image. Step 2: The images (1,3,5) are all distinct. Step 3: In finite sets, distinct images clearly show one-one property.

Step 1: Assuming (f(a)=f(b)), we get (\frac{a}{a+1}=\frac{b}{b+1}). Step 2: Cross-multiplication gives (a(b+1)=b(a+1)), so (ab+a=ab+b), hence (a=b). Step 3: In rational functions, remove the denominator-zero value from the domain before checking one-one.

Step 1: (\lfloor x\rfloor) gives the greatest integer less than or equal to (x). Step 2: (f(1.2)=1) and (f(1.8)=1), while (1.2\neq1.8). Step 3: Step-like functions can give the same value for many inputs.

Step 1: On the given interval, (\tan x) is strictly increasing. Step 2: Therefore two different (x) values cannot give the same tangent value. Step 3: Restricting a trigonometric function to a suitable interval can make it one-one.

Step 1: Every element of (A) has exactly one image. Step 2: The images (5,8,6,9) are all distinct. Step 3: For finite sets, if no image is repeated, the function is one-one.