One-one function

Correct answer: A. (f) is not one-one

Step 1: This domain contains both positive and negative opposite numbers. Step 2: (f(2)=4) and (f(-2)=4), while (2\neq -2). Step 3: On a symmetric domain (x^2) is not one-one.

Correct answer: A. Because (f(x)=0) for every (x<0)

Step 1: Read the absolute value expression piecewise. Step 2: If (x<0), then (|x|=-x), so (|x|+x=0). Step 3: Many different negative inputs give the same value (0), so the function is not one-one.

Correct answer: A. (f) is one-one

Step 1: Only (\sin x) is periodic, but (x+\sin x) is not periodic. Step 2: As (x) increases, the total value does not return to an earlier value, so different inputs do not share the same value. Step 3: For mixed functions examine the complete rule.

Correct answer: A. (f) is one-one

Step 1: On the given interval (\tan x) is strictly increasing. Step 2: So if \(x_1\neq x_2\), then \(\tan x_1\neq \tan x_2\). Step 3: Taking (\tan x) on its principal interval makes it one-one.

Correct answer: A. One-one

Step 1: (\ln x) increases on its domain ((0,\infty)). Step 2: If (\ln x_1=\ln x_2), then (x_1=x_2). Step 3: A logarithmic function is one-one on its proper domain.

Correct answer: A. (f) is not one-one

Step 1: Even-power functions can give the same power value for opposite inputs. Step 2: (f(1)=2) and (f(-1)=2), while (1\neq -1). Step 3: Check even powers carefully on the whole real domain.

Correct answer: A. (f) is one-one

Step 1: As (x+1) increases, (\sqrt{x+1}) also increases. Step 2: From (\sqrt{x_1+1}=\sqrt{x_2+1}), we get (x_1+1=x_2+1), hence (x_1=x_2). Step 3: A square-root function is one-one on its proper domain.

Correct answer: A. (f) is one-one

Step 1: For (x<0), the values are negative. Step 2: For (x\ge0), the values are (1) or more, so the ranges of the two parts do not overlap. Step 3: For piecewise functions check each part and the overlap of their ranges.

Correct answer: A. (f) is not one-one

Step 1: The ranges of the two parts can overlap. Step 2: (f(-2)=4) and (f(3)=4), while (-2\neq3). Step 3: In a piecewise function same value from different pieces breaks injectivity.

Correct answer: A. (f) is one-one

Step 1: Each input has exactly one image, so it is a function. Step 2: The images (4,5,6) are all distinct. Step 3: For finite sets a function is one-one when images do not repeat.

Correct answer: A. (f) is not one-one

Step 1: Injectivity requires no repetition among images. Step 2: Both (2) and (3) have image (5). Step 3: In ordered-pair questions quickly check the second components.

Correct answer: A. One-one

Step 1: This is a linear function and the coefficient of (x) is (-2). Step 2: Since (-2\neq0), different (x)-values give different outputs. Step 3: A linear function with negative slope can also be one-one.

Correct answer: A. (f) is not one-one

Step 1: A constant function sends every input to the same value. Step 2: (f(1)=7) and (f(2)=7), while (1\neq2). Step 3: A constant function can be one-one only when the domain has a single element.

Correct answer: A. (f(1)=f(3))

Step 1: (x^2-4x+7=(x-2)^2+3). Step 2: (1) and (3) are equally distant from (2), so (f(1)=4) and (f(3)=4). Step 3: For quadratics check points equally distant from the vertex.

Correct answer: A. When (k\neq2)

Step 1: This is a linear function and the coefficient of (x) is (k-2). Step 2: For one-one nature this coefficient must not be zero. Step 3: Hence (k-2\neq0), so (k\neq2).

Correct answer: A. (f) is one-one

Step 1: In (n+2), different (n)-values give different values. Step 2: If (n_1+2=n_2+2), then (n_1=n_2). Step 3: A simple shift function on natural numbers is one-one.

Correct answer: A. Because (f(2)=f(-2))

Step 1: Integers include both (2) and (-2). Step 2: (2^2=4) and ((-2)^2=4), while (2\neq-2). Step 3: Even-power rules are not one-one on integers either.

Correct answer: A. (f) is one-one

Step 1: (2n+1) is a linear rule. Step 2: If (2n_1+1=2n_2+1), then (n_1=n_2). Step 3: A linear function with non-zero coefficient is one-one on integers too.

Correct answer: A. No

Step 1: On \(-1\le x\le1\), this sum becomes constant. Step 2: For example \(f(0)=2\) and \(f\left(\frac{1}{2}\right)=2\), while \(0\neq \frac{1}{2}\). Step 3: In sums of absolute values look for constant intervals.

Correct answer: A. (f(0)=f(-3))

Step 1: It is enough to find two different inputs with the same value. Step 2: (f(0)=0) and (f(-3)=-27+27=0), while (0\neq-3). Step 3: For polynomials use factors or special values to test quickly.

Correct answer: A. (f) is one-one

Step 1: (x^2-4x+5=(x-2)^2+1). Step 2: The domain (x\ge2) is to the right of the vertex where the function increases. Step 3: A quadratic restricted to one side of the vertex is one-one.

Correct answer: A. Every horizontal line cuts the graph at at most one point

Step 1: Injectivity means the same (y)-value must not occur at two different (x)-values. Step 2: A horizontal line represents a fixed (y)-value. Step 3: Hence the horizontal line test is a quick way to check one-one nature.

Correct answer: A. (f(p)=2,\ f(q)=3,\ f(r)=2)

Step 1: A one-one function needs three distinct images for three distinct inputs. Step 2: In the first option both (p) and (r) have image (2). Step 3: In such options the fastest method is to look for repeated images.

Correct answer: A. (f) is one-one

Step 1: This function is in linear form. Step 2: From (\frac{2x_1+3}{5}=\frac{2x_2+3}{5}), we get (2x_1+3=2x_2+3), hence (x_1=x_2). Step 3: Such a function is one-one when the coefficient of (x) in the numerator is non-zero.

Correct answer: A. (k\ge0)

Step 1: For injectivity it is useful to check whether the function keeps moving in one direction. Step 2: When (k\ge0), both (x^3) and (kx) contribute an increasing effect, so different (x)-values give different outputs. Step 3: In parameter-based questions always check how the parameter changes increasing or decreasing behavior.