One-one function

Step 1: Check monotonic behaviour to test one-one nature. Step 2: (f'(x)=3x^2-3=3(x-1)(x+1)), so the function is increasing on ((-\infty,-1]). Step 3: In exams, the sign of the derivative is the quickest method for such cubic functions.

Step 1: Write (f(x)=x^2+4x+7=(x+2)^2+3). Step 2: The vertex is at (x=-2), and the function is strictly increasing to the right of it. Step 3: A quadratic function is not one-one on all real numbers, but it is one-one on one side of its vertex.

Step 1: In a linear function, the slope is (a). Step 2: If (a\ne 0), distinct (x)-values give distinct (f(x))-values. Step 3: A constant function is not one-one, so (a=0) must be avoided.

Step 1: In a one-one function, distinct elements of (A) must get distinct images in (B). Step 2: The first element has (4) choices, the second has (3), and the third has (2), so total (4\cdot3\cdot2=24). Step 3: Remember that order matters in function assignment.

Step 1: For a one-one function, the codomain must have at least as many elements as the domain. Step 2: Here (A) has (5) elements and (B) has only (3), so distinct images are impossible. Step 3: If the domain is larger than the codomain, the number of one-one functions is (0).

Step 1: Use the derivative to study monotonicity. Step 2: (f'(x)=e^x+1), which is positive for every (x). Hence the function is strictly increasing and one-one. Step 3: A function with derivative always positive or always negative is one-one.

Step 1: (\sin x) is periodic on the real line, so it is not one-one on all real numbers. Step 2: On ([-\frac{\pi}{2},\frac{\pi}{2}]), (\sin x) is strictly increasing. Step 3: For trigonometric functions, choose a proper restricted interval to make them one-one.

Step 1: The domain of (\ln x) is ((0,\infty)). Step 2: (f'(x)=\frac{1}{x}), which is positive throughout this domain, so the function is increasing. Step 3: An increasing logarithmic function is one-one on its domain.

Step 1: The graph of (|x-2|) turns at (x=2). Step 2: On ([2,\infty)), it becomes (x-2) and is strictly increasing. Step 3: To make a modulus function one-one, restrict the domain to one side of the turning point.

Step 1: Find (f'(x)=5x^4+1). Step 2: (5x^4+1>0) for every real (x), so (f) is strictly increasing. Step 3: A strictly increasing function is one-one.

Step 1: For one-one nature, different (x)-values must have different images. Step 2: (f(1)=2) and (f(-1)=2), while (1\ne -1). Step 3: Once two distinct inputs give the same output, the function is not one-one.

Step 1: For finite sets of equal size, a one-one function acts like a permutation. Step 2: The (4) elements can be mapped distinctly in (4!=24) ways. Step 3: When domain and codomain have equal size (n), the number of one-one functions is (n!).

Step 1: To prove not one-one, it is enough to find two different inputs with the same output. Step 2: (f(2)=\frac{2}{5}) and (f(\frac{1}{2})=\frac{\frac{1}{2}}{\frac{1}{4}+1}=\frac{2}{5}). Step 3: In such questions, finding a smart pair can be faster than using derivatives.

Step 1: Put (f(a)=f(b)), giving (3a-7=3b-7). Step 2: This gives (3a=3b), hence (a=b). Step 3: If (f(a)=f(b)\Rightarrow a=b), the function is one-one.

Step 1: Test the function at two different values. Step 2: (f(0)=0) and (f(-1)=(-1)^3+(-1)^2=0), while (0\ne -1). Step 3: One pair of distinct inputs with the same image breaks one-one nature on the whole domain.

Step 1: Each domain element has exactly one image, so it is a function. Step 2: Both (1) and (3) have image (5), so distinct inputs do not have distinct images. Step 3: While checking one-one nature, focus on repeated images.

Step 1: Assume (f(a)=f(b)). Step 2: (\frac{1}{a}=\frac{1}{b}) gives (a=b), since (a,b>0). Step 3: For reciprocal-type functions, direct proof using equal images is very effective.

Step 1: (\tan x) is periodic and is not defined everywhere. Step 2: On ((-\frac{\pi}{2},\frac{\pi}{2})), it is strictly increasing and takes each value once. Step 3: Remember standard one-one intervals for inverse trigonometric functions.

Step 1: Both (1) and (-1) are present in integers. Step 2: (f(1)=1) and (f(-1)=1), but (1\ne -1). Step 3: The square function is symmetric, so it is generally not one-one on a large symmetric domain.

Step 1: (f(a)=f(b)\Rightarrow a+2=b+2\Rightarrow a=b), so the function is one-one. Step 2: Values like (1) and (2) in (\mathbb{N}) are not images of any (n), so it is not onto. Step 3: Always test one-one and onto separately.

Step 1: Since (\cos x) is periodic, it is not one-one on all (\mathbb{R}). Step 2: On ([0,\pi]), (\cos x) is strictly decreasing. Step 3: A strictly increasing or decreasing function is one-one on the chosen interval.

Step 1: Assume (f(a)=f(b)). Step 2: (a^3=b^3) gives (a=b), because the cube function preserves order on real numbers. Step 3: Simple odd-power functions are often one-one on the whole real domain.

Step 1: Write (x^2-6x+10=(x-3)^2+1). Step 2: The vertex is at (x=3), and the function is strictly increasing on ([3,\infty)). Step 3: For a quadratic function, choose one side of the vertex.

Step 1: The (5) elements of (A) must be mapped to (5) distinct elements of (B). Step 2: The number of choices is (6\cdot5\cdot4\cdot3\cdot2=720). Step 3: Use (^{m}P_n) for counting one-one functions from an (n)-element domain to an (m)-element codomain.

Step 1: (f'(x)=1+\cos x). Step 2: (1+\cos x\ge 0) for every (x), and the function does not repeat the same value at two distinct points. Step 3: In such questions, combine derivative sign with overall increasing behaviour.