Functions

Correct answer: A. (g=f^{-1})

Step 1: From (y=3x-2), we get (x=\frac{y+2}{3}). Step 2: This matches the form of (g(y)), so (g) is the inverse function. Step 3: You can also verify an inverse by checking (f(g(x))=x) or (g(f(x))=x).

Correct answer: B. (24)

Step 1: The first element has (4) choices, the second has (3), and the third has (2). Step 2: Total one-one functions are (4\cdot3\cdot2=24). Step 3: For one-one functions, choose images without repetition.

Correct answer: B. It is neither one-one nor onto

Step 1: (\sin 0=\sin 2\pi), so the function is not one-one. Step 2: The range of (\sin x) is ([-1,1]), so it does not cover all of (\mathbb{R}). Step 3: For trigonometric functions, check periodicity and range.

Correct answer: A. Both one-one and onto

Step 1: On ([0,\infty)), (x^2) is increasing, so it is one-one. Step 2: For every (y\geq0), (x=\sqrt{y}) lies in the domain. Step 3: Restricting the domain can make (x^2) bijective.

Correct answer: A. ([-1,\infty))

Step 1: Write (x^2-2x=(x-1)^2-1). Step 2: Since ((x-1)^2\geq0), the least value is (-1). Step 3: An upward-opening quadratic has range starting from its minimum value.

Correct answer: A. (\mathbb{R}\setminus{-1})

Step 1: The denominator of a fraction must not be zero. Step 2: (x+1=0) gives (x=-1). Step 3: To find the domain, first remove values that make the denominator zero.

Correct answer: C. It is constant and not one-one

Step 1: For every (x), the function value is (5). Step 2: Different inputs give the same output, so it is not one-one. Step 3: A constant function gives only one value, so it is usually not onto either.

Correct answer: B. ((f\circ g)(x)=2x+1) and ((g\circ f)(x)=2x+2)

Step 1: (f(g(x))=f(2x)=2x+1). Step 2: (g(f(x))=g(x+1)=2x+2). Step 3: Composition of functions is not generally commutative.

Correct answer: A. ((-1,1))

Step 1: For (x\geq0), the value is (\frac{x}{1+x}), which approaches (1) but never reaches it. Step 2: For (x<0), the value lies between (-1) and (0). Step 3: Combining both parts gives the range ((-1,1)).

Correct answer: A. (\sqrt[3]{\frac{x+1}{2}})

Step 1: Put (y=2x^3-1). Step 2: Then (y+1=2x^3), so (x=\sqrt[3]{\frac{y+1}{2}}). Step 3: In inverse questions, isolate (x) first and then change the variable.

Correct answer: A. Because (1) has two preimages (1) and (-1)

Step 1: For the inverse to be a function, the original function must be one-one. Step 2: (f(1)=f(-1)=1), so one output has two inputs. Step 3: Thus the inverse relation assigns two values and is not a function.

Correct answer: A. ([4,\infty))

Step 1: (x^2\geq0) for every real (x). Step 2: Hence (x^2+4\geq4), and (4) is attained at (x=0). Step 3: While writing range, check whether the endpoint is included.

Correct answer: A. Because (2) has no preimage

Step 1: The range of (\cos x) is ([-1,1]). Step 2: The codomain (\mathbb{R}) contains (2), but (\cos x=2) is impossible. Step 3: To test onto, look for values in the codomain that are missed.

Correct answer: A. Because (f(0)=f(\sqrt{3}))

Step 1: To show a function is not one-one, find two different inputs with the same output. Step 2: (f(0)=0) and (f(\sqrt{3})=(\sqrt{3})^3-3\sqrt{3}=0). Step 3: A suitable counterexample is enough to disprove one-one behaviour.

Correct answer: A. Both one-one and onto

Step 1: (e^x) is strictly increasing on the real line, so it is one-one. Step 2: Its values are positive, and every positive number can be written as (e^x). Step 3: With codomain ((0,\infty)), it becomes bijective.

Correct answer: A. One-one but not onto

Step 1: (e^x) is increasing, so it gives different outputs for different inputs. Step 2: It never becomes zero or negative, so it does not cover all of (\mathbb{R}). Step 3: The same formula can have different properties with a different codomain.

Correct answer: A. Because (\tan x) is not defined for all real (x)

Step 1: (\tan x) is not defined where (\cos x=0). Step 2: For example, at (x=\frac{\pi}{2}), (\tan x) is not defined. Step 3: A function must assign a value to every element of its domain.

Correct answer: C. (3)

Step 1: Let (y=\frac{3x+1}{x+2}). Step 2: Putting (y=3) gives (3x+1=3x+6), which is impossible. Step 3: In linear fractional functions, the horizontal limiting value is often the missed value.

Correct answer: A. It is bijective

Step 1: The images of the three elements of (A) are all distinct. Step 2: The outputs include (1,2,3), so the function is onto. Step 3: For finite equal sets, distinct images imply bijection.

Correct answer: A. (0)

Step 1: An absolute value is never negative. Step 2: At (x=2), (|x-2|=0). Step 3: The minimum of an absolute value function occurs when the inside expression becomes zero.

Correct answer: C. It is neither one-one nor onto

Step 1: (\lfloor 1.2\rfloor=\lfloor 1.8\rfloor=1), so it is not one-one. Step 2: Its values are always integers, so real values like (0.5) are missed. Step 3: For step functions, test one-one and onto separately.

Correct answer: A. No real (a) is possible

Step 1: For any real (a), this is a quadratic function. Step 2: On all of (\mathbb{R}), a quadratic takes equal values on opposite sides of its vertex. Step 3: Therefore it cannot be one-one on the entire real domain.

Correct answer: A. It is always one-one

Step 1: Suppose ((g\circ f)(x_1)=(g\circ f)(x_2)). Step 2: Since (g) is one-one, (f(x_1)=f(x_2)); since (f) is one-one, (x_1=x_2). Step 3: The composition of one-one functions is one-one.

Correct answer: A. It is always onto

Step 1: Take any (c\in C). Step 2: Since (g) is onto, some (b\in B) satisfies (g(b)=c); since (f) is onto, some (a\in A) satisfies (f(a)=b). Step 3: Hence ((g\circ f)(a)=c), so the composition is onto.

Correct answer: A. Domain (\mathbb{R}\setminus{1}), missed value (1)

Step 1: The denominator is (x-1), so (x=1) is removed from the domain. Step 2: Putting (y=1) in (y=\frac{x+1}{x-1}) gives (x+1=x-1), impossible. Step 3: For rational functions, check domain and missed range values separately.