Onto function

Step 1: Put (f(x)=y), giving (x^3+1=y). Step 2: This gives (x=\sqrt[3]{y-1}), which is real for every real (y). Step 3: A cubic form can obtain all real values.

Step 1: (x^2\ge0), so (x^2+2\ge2). Step 2: For every (y\ge2), take (x=\sqrt{y-2}), then (f(x)=y). Step 3: The range ([2,\infty)) equals the codomain.

Step 1: The minimum value of (x^2+2) is (2). Step 2: The codomain (\mathbb{R}) contains (1), but (x^2+2=1) has no real solution. Step 3: Minimum or maximum values help identify the range quickly.

Step 1: (x^2\ge0), so (4-x^2\le4). Step 2: For every (y\le4), take (x=\sqrt{4-y}), then (f(x)=y). Step 3: A downward quadratic can be onto ((-\infty,4]).

Step 1: The maximum value of (4-x^2) is (4). Step 2: The codomain (\mathbb{R}) contains (5), but (4-x^2=5) has no real solution. Step 3: Codomain values above the maximum are missed.

Step 1: Take any codomain value (y\ne0). Step 2: Put (x=\frac{1}{y}), then (x\ne0) and (f(x)=y). Step 3: The reciprocal function is onto with the correct domain and codomain.

Step 1: (\frac{1}{x}) is never equal to (0). Step 2: The codomain (\mathbb{R}) contains (0), but it is not in the range. Step 3: For reciprocal functions, check whether (0) is included in the codomain.

Step 1: The minimum value of (x^2+1) is (1). Step 2: The codomain ((0,\infty)) contains (\frac{1}{2}), but this value cannot be obtained. Step 3: If the codomain contains values below the minimum range, the function is not onto.

Step 1: The range of (x^2+1) is ([1,\infty)). Step 2: The codomain is also ([1,\infty)), so every codomain value is obtained. Step 3: When the codomain equals the range, the function is onto.

Step 1: For every real (y), (x=\sqrt[5]{y}) is real. Step 2: For this (x), (x^5=y). Step 3: Simple odd-power functions are onto from (\mathbb{R}) to (\mathbb{R}).

Step 1: (x^4) is always (0) or positive. Step 2: The codomain (\mathbb{R}) contains negative values such as (-1), but they are not in the range. Step 3: Even-power functions do not give negative values on real inputs.

Step 1: The range of (x^4) is ([0,\infty)). Step 2: For every (y\ge0), take (x=\sqrt[4]{y}), then (x^4=y). Step 3: An even-power function can be onto a non-negative codomain.

Step 1: Solve (5x=y). Step 2: This gives (x=\frac{y}{5}), which is real for every real (y). Step 3: A function of the form (ax) with (a\ne0) is onto on (\mathbb{R}).

Step 1: (0x+4=4) for every (x). Step 2: Hence the range is only ({4}), while the codomain is (\mathbb{R}). Step 3: A constant function is not onto a large codomain.

Step 1: (|x|\ge0), so (|x|+3\ge3). Step 2: For every (y\ge3), take (x=y-3), then (|x|+3=y). Step 3: A vertical shift of a modulus function shifts its range upward.

Step 1: The minimum value of (|x|+3) is (3). Step 2: The codomain (\mathbb{R}) contains (0), but this value cannot be obtained. Step 3: Values below the minimum are missed from the range.

Step 1: For any (y\in\mathbb{R}), put (x^3-2=y). Step 2: This gives (x=\sqrt[3]{y+2}), valid for every real (y). Step 3: A vertical shift of a cubic function remains onto (\mathbb{R}).

Step 1: (2n-1) always gives an odd natural number. Step 2: The codomain (\mathbb{N}) contains even numbers such as (2), which are not in the range. Step 3: For natural-number functions, separate even and odd outputs.

Step 1: The codomain consists only of odd natural numbers. Step 2: Every odd number (2k-1) is obtained by taking (n=k). Step 3: Choosing the codomain correctly can make the function onto.

Step 1: The range of (\sin x) is ([-1,1]). Step 2: Therefore the range of (2\sin x) is ([-2,2]), equal to the codomain. Step 3: Multiplying a trigonometric function scales its range.

Step 1: The range of (2\sin x) is ([-2,2]). Step 2: Every value in the codomain ([-1,1]) is obtained for some (x). Step 3: For onto nature, every codomain value must be hit; however, the function must also be well-defined with outputs in the codomain.

Step 1: The value of (\frac{x}{1+|x|}) always lies between (-1) and (1). Step 2: It can never be equal to (1) or (-1), so these codomain values are missed. Step 3: Always check endpoint values in bounded-range questions.

Step 1: The range of this function is ((-1,1)). Step 2: The codomain is also ((-1,1)), so every codomain value is obtained. Step 3: In an open interval, endpoint values are not required.

Step 1: (f'(x)=3x^2+1>0), so the function is increasing. Step 2: As (x\to\infty), (f(x)\to\infty), and as (x\to-\infty), (f(x)\to-\infty). Step 3: Such an increasing continuous function is onto (\mathbb{R}).

Step 1: Onto means every element of the codomain is in the range. Step 2: Not onto means at least one codomain element is missed. Step 3: Understanding the negation of the definition is very useful in exams.