One-one function

Step 1: (x^2+2x=(x+1)^2-1). Step 2: (f(0)=0) and (f(-2)=0), while (0\neq -2). Step 3: In a quadratic function, test inputs equally distant from the vertex.

Step 1: For testing one-one, assume (f(a)=f(b)). Step 2: From (9a=9b), we get (a=b). Step 3: A simple linear expression with a non-zero multiplier is one-one.

Step 1: Each element of the domain has one image. Step 2: The images (4), (6), and (7) are all distinct. Step 3: Some codomain elements may be unused without affecting one-one nature.

Step 1: (0x+8=8), so it is a constant function. Step 2: (f(1)=8) and (f(2)=8), while (1\neq 2). Step 3: In a linear form, if the coefficient of (x) is (0), the function is not one-one.

Step 1: The cube root is defined for every real number. Step 2: If (\sqrt[3]{a}=\sqrt[3]{b}), cubing both sides gives (a=b). Step 3: The cube-root function is increasing, so it is one-one.

Step 1: In this domain, (x-2\geq 0). Step 2: If ((a-2)^2=(b-2)^2) and (a,b\geq 2), then (a-2=b-2), so (a=b). Step 3: A quadratic can be one-one when restricted to one side of its vertex.

Step 1: The center of ((x+5)^2) is (x=-5). Step 2: (f(-4)=1) and (f(-6)=1), but (-4\neq -6). Step 3: In a perfect square, values equally distant from the center quickly give the answer.

Step 1: (x^5) is an odd-power function and increases on (\mathbb{R}). Step 2: If (a^5=b^5), then (a=b). Step 3: Basic odd-power functions are generally one-one on real numbers.

Step 1: Check values at two different inputs. Step 2: (f(0)=0) and (f(-1)=0), while (0\neq -1). Step 3: A quadratic function on all real numbers is often not one-one.

Step 1: Writing (f(a)=f(b)) gives (4a+4=4b+4). Step 2: Subtracting (4) and dividing by (4) gives (a=b). Step 3: This is the standard method to prove one-one nature.

Step 1: In (|2x|), (x) and (-x) still give the same value. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq -1). Step 3: A multiplier inside modulus does not make it one-one on all real numbers.

Step 1: (\sqrt{x}) is increasing on its domain ([0,\infty)). Step 2: If (\sqrt{a}=\sqrt{b}), squaring gives (a=b). Step 3: An increasing function gives different values for different inputs.

Step 1: (\sqrt{x^2}=|x|). Step 2: (f(2)=2) and (f(-2)=2), while (2\neq -2). Step 3: Treat (\sqrt{x^2}) as modulus while solving.

Step 1: (f(x)=x) sends every element to itself. Step 2: Different elements keep different images. Step 3: The identity function is always one-one.

Step 1: The center of the modulus is (x=-1). Step 2: (f(0)=4) and (f(-2)=4), while (0\neq -2). Step 3: Adding a constant outside does not remove equal-image pairs.

Step 1: (x^2-2x+1=(x-1)^2). Step 2: Since the domain is (x\geq 1), (x-1\geq 0), so the square increases. Step 3: A quadratic is one-one on one side of its vertex.

Step 1: (f(x)=(x-1)^2). Step 2: (f(0)=1) and (f(2)=1), while (0\neq 2). Step 3: The same formula can have a different one-one result on a different domain.

Step 1: Assume (f(a)=f(b)). Step 2: From (\frac{3a-1}{5}=\frac{3b-1}{5}), we get (3a-1=3b-1), so (a=b). Step 3: A non-zero constant denominator does not change the one-one nature of a linear function.

Step 1: (x^6) is an even power, so opposite inputs give equal values. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq -1). Step 3: Adding a constant to an even-power function does not make it one-one.

Step 1: Consider the function (x^3+x). Step 2: It is strictly increasing on (\mathbb{R}) because both terms increase with (x). Step 3: A strictly increasing function is one-one, so different inputs cannot give the same value.

Step 1: Test values at two different inputs. Step 2: (f(-1)=0), (f(0)=0), and (-1\neq 0). Step 3: Not every cubic expression is automatically one-one, so checking examples is important.

Step 1: In (|x|), (x) and (-x) have the same value. Step 2: (f(3)=7) and (f(-3)=7), while (3\neq -3). Step 3: Multiplying or adding outside the modulus does not remove equal-image pairs.

Step 1: Find the images of the domain elements. Step 2: (f(2)=1), (f(4)=2), (f(6)=3), and all images are distinct. Step 3: For a finite domain, a direct value table is a quick test for one-one nature.

Step 1: Check remainders for even and odd numbers. Step 2: (f(1)=1) and (f(3)=1), while (1\neq 3). Step 3: Remainder-based functions often send many inputs to the same value, so test one-one nature carefully.

Step 1: The exponential function (e^x) increases continuously as (x) increases. Step 2: If (e^a=e^b), then the same base gives (a=b). Step 3: In exams, a strictly increasing function gives different images for different inputs, so it is one-one.