One-one function

Step 1: (x^2) has the same value at (x) and (-x). Step 2: Hence (f(1)=e^1) and (f(-1)=e^1), while (1\ne-1). Step 3: Even with an exponential outer function, an even inner expression can break one-one nature.

Step 1: On ((0,\infty)), (x^2) is strictly increasing. Step 2: (\ln u) is also strictly increasing for (u>0), so (\ln(x^2)) is one-one on this domain. Step 3: The one-one nature of the same formula can change when the domain changes.

Step 1: Both (2) and (-2) are included in the domain. Step 2: (f(2)=\ln4) and (f(-2)=\ln4), while (2\ne-2). Step 3: When a domain contains opposite values, check expressions involving (x^2).

Step 1: The image of (1) is fixed as (1). Step 2: The remaining (4) elements can be mapped one-one to the remaining (4) elements in (4!=24) ways. Step 3: For equal-size sets, one-one functions are counted like permutations.

Step 1: Since the function is one-one, the (7) elements of (A) have (7) distinct images. Step 2: (B) also has exactly (7) elements, so all elements of (B) are covered. Step 3: For finite sets of equal size, a one-one function is also onto.

Step 1: If (x>0), then (x^3+x\cos^2x>0), and if (x<0), the value is negative. Step 2: The function has odd-type behaviour and preserves the direction of values with the sign of (x). Step 3: For difficult forms, combine sign behaviour with increasing tendency to judge one-one nature.

Step 1: (f'(x)=3x^2+\cos x). Step 2: For large (|x|), the (3x^2) part strongly supports increasing behaviour, and the expression is not a repeating periodic form. Step 3: A small trigonometric part does not always destroy one-one nature when a dominant cubic part controls the function.

Step 1: (f'(x)=1+\sin x). Step 2: (1+\sin x\ge0) for every real (x), and the function does not turn back. Step 3: For such functions, a non-negative derivative along with overall increasing behaviour indicates one-one nature.

Step 1: (f'(x)=1-\sin x). Step 2: (1-\sin x\ge0) for every (x), and the function moves overall in the increasing direction. Step 3: A derivative may become zero at points, yet the function can still be one-one if it does not repeat values.

Step 1: For (x<0), (\max{x,0}=0). Step 2: Thus distinct inputs such as (-1) and (-2) both have image (0). Step 3: In piecewise-defined functions, first check any constant part.

Step 1: For (x\ge1), (\min{x,1}=1). Step 2: Hence (f(2)=1) and (f(3)=1), while (2\ne3). Step 3: A function that is constant on a ray cannot be one-one.

Step 1: The images of the four domain elements are (2,4,1,3). Step 2: All of them are distinct, so no two different inputs have the same image. Step 3: For finite sets, list the images to check one-one nature quickly.

Step 1: Each domain element has an image, so it is a function. Step 2: Both (1) and (2) have image (3), so distinct inputs do not have distinct images. Step 3: Repeated images destroy one-one nature.

Step 1: The function contains (x^2), so opposite (x)-values may give equal outputs. Step 2: (f(1)=\frac{1}{2}) and (f(-1)=\frac{1}{2}), while (1\ne-1). Step 3: For even-form rational functions, always test (x) and (-x).

Step 1: (f'(x)=\frac{(x-1)-(x+2)}{(x-1)^2}=\frac{-3}{(x-1)^2}). Step 2: This is always negative for (x>1), so the function is strictly decreasing. Step 3: A strictly decreasing function is also one-one.

Step 1: (x^6) is an even power, so it gives the same value at (x) and (-x). Step 2: (f(1)=2) and (f(-1)=2), while (1\ne-1). Step 3: Do not assume an even-power polynomial is one-one on the whole real domain.

Step 1: The domain ([0,\infty)) has no negative values. Step 2: On this domain, as (x) increases, (x^6+1) increases, so the same value is not obtained at two different (x)-values. Step 3: An even-power function can become one-one when the domain is suitably restricted.

Step 1: (x^3-3x^2+3x=(x-1)^3+1). Step 2: ((x-1)^3+1) is strictly increasing on all of (\mathbb{R}). Step 3: Recognizing a perfect cubic form saves time in exams.

Step 1: Try simple values to find repeated images. Step 2: (f(0)=0) and (f(3)=27-27=0), while (0\ne3). Step 3: One pair of different inputs with the same image is enough to reject one-one nature.

Step 1: The cube-root function is defined for every real number. Step 2: As (x) increases, (\sqrt[3]{x}) also increases, so two different inputs do not give the same value. Step 3: The cube-root function is one-one on the whole real domain.

Step 1: Because of (x^2), the inside value is the same at (x) and (-x). Step 2: (f(1)=\sqrt[3]{1}=1) and (f(-1)=\sqrt[3]{1}=1), while (1\ne-1). Step 3: Even with a cube-root outer form, the square inside breaks one-one nature.

Step 1: (f) sends distinct inputs of (A) to distinct images. Step 2: (S) is only a smaller part of (A), so distinct elements of (S) also keep distinct images. Step 3: A restriction of a one-one function is also one-one.

Step 1: (f(x)=(x+1)^2+1), so the vertex is at (x=-1). Step 2: On ((-\infty,-1]), the function is strictly decreasing and hence one-one. Step 3: Remember that a quadratic decreases to the left of its vertex and increases to the right.

Step 1: (f'(x)=3ax^2+b). Step 2: Since (a>0) and (b>0), (3ax^2+b>0) for every real (x). Step 3: If the derivative is always positive, the function is strictly increasing and one-one.

Step 1: Differentiating gives (f'(x)=\frac{x^2(x^2+3)}{(1+x^2)^2}). Step 2: This is non-negative for every real (x), and the function does not repeat values, so it is one-one. Step 3: For rational functions, checking the sign of the derivative is a reliable method.