Which three of the twelve basic functions are bounded( above and below)?

Three of the tweleve basic functions are even. Which are they?

Ah, functions and boundaries, the perfect opportunity for some mathematical clowning around!

Three of the twelve basic functions that are bounded above and below are:

1. The constant function: It's like that friend who always stays in the same place. No matter how much you try to push or pull, it remains bounded and unyielding.

2. Sine function: This wavy character oscillates up and down but never strays too far from the x-axis. It loves to boogie but has its limits, just like all of us on the dance floor.

3. Hyperbolic tangent function: You can think of this one as a roller coaster of function. It goes up and down, twists and twirls, but never ventures too far away from its upper and lower bounds. It's like a thrill ride that keeps things in check.

As for the three basic functions that are even, they are:

1. Cosine function: This function is as even as they come, just like those well-organized shelves that have everything neatly balanced. It's symmetry at its finest.

2. Secant function: This reciprocal of the cosine function is also even, just like that friend who always brings chocolate to share. It's always seeking balance and fairness.

3. Cosecant function: Another reciprocal function, the cosecant, is even too! It's like that friend who always makes sure there are equal amounts of silly jokes and laughter in every conversation.

Remember, math can be funny too, so be sure to keep smiling, solving, and clowning around!

The three of the twelve basic functions that are bounded above and below are:

1. Polynomial functions: Polynomial functions are bounded above and below if their degree is even. For example, the function f(x) = x^2 is a polynomial function that is bounded below by 0.

2. Exponential functions: Exponential functions of the form f(x) = a^x, where a is a positive constant, are bounded above and below. For example, the function f(x) = 2^x is an exponential function that is bounded below by 0.

3. Trigonometric functions: Trigonometric functions, such as sine (sin(x)), cosine (cos(x)), and tangent (tan(x)), are bounded above and below. For example, the function f(x) = sin(x) is a trigonometric function that is bounded between -1 and 1.

The three of the twelve basic functions that are even (symmetric about the y-axis) are:

1. Polynomial functions: Even degree polynomial functions, such as f(x) = x^2, f(x) = x^4, etc., are symmetric about the y-axis.

2. Trigonometric functions: The cosine function, cos(x), is even and symmetric about the y-axis.

3. Exponential functions: Exponential functions with an even exponent, such as f(x) = (2^x)^2 = 2^(2x), are even and symmetric about the y-axis.

To determine which three of the twelve basic functions are bounded above and below, we need to understand the concept of boundedness for functions. A function is bounded above if there exists a number such that the function value never exceeds this number. Similarly, a function is bounded below if there exists a number such that the function value never falls below this number.

The twelve basic functions usually referred to are:

1. Constant function (f(x) = c)
2. Identity function (f(x) = x)
3. Squaring function (f(x) = x^2)
4. Cube function (f(x) = x^3)
5. Absolute value function (f(x) = |x|)
6. Exponential function (f(x) = e^x)
7. Logarithmic function (f(x) = log(x))
8. Sine function (f(x) = sin(x))
9. Cosine function (f(x) = cos(x))
10. Tangent function (f(x) = tan(x))
11. Square root function (f(x) = √x)
12. Reciprocal function (f(x) = 1/x)

Now, let's determine which three functions among these are bounded above and below.

1. The constant function (f(x) = c) is bounded both above and below because its value remains the same for all x.

2. The identity function (f(x) = x) is not bounded above or below. It extends to infinity in both directions.

3. The squaring function (f(x) = x^2), cube function (f(x) = x^3), exponential function (f(x) = e^x), logarithmic function (f(x) = log(x)), and reciprocal function (f(x) = 1/x) are not bounded above or below. They extend to infinity in at least one direction.

4. The absolute value function (f(x) = |x|) and tangent function (f(x) = tan(x)) are bounded below but not above. They do not have a maximum value but have a minimum value for certain intervals.

5. The sine function (f(x) = sin(x)) and cosine function (f(x) = cos(x)) are bounded above and below. The maximum value of sine is 1, and the minimum value is -1. Similarly, the maximum value of cosine is 1, and the minimum value is -1.

6. The square root function (f(x) = √x) is bounded below but not above. It does not have a maximum value but has a minimum value for non-negative inputs (x ≥ 0).

Based on these explanations, the three basic functions that are bounded above and below are:

1. The constant function (f(x) = c)
2. The sine function (f(x) = sin(x))
3. The cosine function (f(x) = cos(x)

Moving on to the second question, we need to identify the three even functions among the twelve basic functions.

An even function is a function that satisfies the property f(x) = f(-x), meaning the function's output remains the same when the input is replaced by the opposite value.

Among the twelve basic functions:

1. The constant function (f(x) = c) is considered even because replacing x with -x keeps the function value unchanged.
2. The squaring function (f(x) = x^2) is also even because (-x)^2 gives the same function value as x^2.
3. The cosine function (f(x) = cos(x)) is even because cos(-x) gives the same function value as cos(x).

Therefore, the three basic functions that are even are:

1. The constant function (f(x) = c)
2. The squaring function (f(x) = x^2)
3. The cosine function (f(x) = cos(x))

Are you referring to the trigonometric functions? Namely sin, cos, tan, csc, sec, and cot that make up the first 6, and their inverses that make the other 6?

It would be a good exercise to sketch each of the three basic functions (sin, cos, and tan) to familiarize yourself with the nature of these functions, if this was not already done in class by your teacher.

The functions csc, sec and cot are the reciprocals. The reciprocals go to zero whenever x approaches 0.

The graphs of inverses poses an extra challenge. Since inverses of periodic functions exist for a particular interval, so the sin, cos and tan functions have to be reduced to a monotonic function that covers the range. For example, for sin(x), the interval chosen is -π/2 to π/2. For cosine, it is from 0 to π. For tangent, the interval chosen is from -π/2 to π/2.
From there, you can deduce the graph of the inverses by reflection about the line y=x.

If you use the textbook Precalculus by Stewart,Redlin and Watson, you can find a lot of information in Sect. 7.4 between pages 550-557.