The Identity Function
The Squaring Function
The Cubing Function
The Reciprocal Function
The Square Root Function
The Exponential Functional Lo
The Natural Logarithum Function
The Sine Function
The Cosine Function
The Absolute Value Function
The Greatest Integer Function
The Logistic Function
Only three of the twelve basic functions are bounded (abover and below). Which three?
Three of the twelve basic function are even. Which are they?
What you need to find out is what each function does, and consequently the range of each of the functions when x varies in the range (-∞infin;).
For your information, the identity function is f(x) = x, thus when x=∞ f(x)=∞ also.
You already know about the sine(x) and cosine(x) functions from trigonometry.
The description of the logistic function can be found here:
http://en.wikipedia.org/wiki/Logistic_function
The reciprocal function has a vertical asymptote at x=0.
If you have problems locating the three functions, post your thinking.
To identify which three of the twelve basic functions are bounded, we need to understand what it means for a function to be bounded. A function is said to be bounded if there exist real numbers M and N such that for every input x, the output values of the function are always between M and N.
Now, let's consider each of the twelve basic functions:
1. The Identity Function (f(x) = x): This function is not bounded, as it continues to increase indefinitely for both positive and negative values of x.
2. The Squaring Function (f(x) = x^2): This function is not bounded either, as it grows without limit as x approaches positive or negative infinity.
3. The Cubing Function (f(x) = x^3): Similar to the previous two functions, the cubing function is not bounded.
4. The Reciprocal Function (f(x) = 1/x): This function is bounded, as it has an upper boundary of 0 and lower boundary of negative infinity.
5. The Square Root Function (f(x) = √x): This function is bounded, as it has an upper boundary of positive infinity, but it does not have a lower boundary.
6. The Exponential Function (f(x) = e^x): This function is not bounded, as it grows rapidly as x approaches positive infinity.
7. The Natural Logarithm Function (f(x) = ln(x)): Similar to the reciprocal function, the natural logarithm function is bounded, but with an upper boundary of positive infinity.
8. The Sine Function (f(x) = sin(x)): The sine function is not bounded, as it oscillates between -1 and 1 infinitely.
9. The Cosine Function (f(x) = cos(x)): Similar to the sine function, the cosine function is also not bounded.
10. The Absolute Value Function (f(x) = |x|): This function is bounded, as it has an upper and lower boundary of positive infinity and 0 respectively.
11. The Greatest Integer Function (f(x) = ⌊x⌋): This function is not bounded, as it "jumps" at every integer value.
12. The Logistic Function (f(x) = L / (1 + e^(-kx))): The logistic function is bounded, as it has an upper and lower boundary of L and 0 respectively.
Considering the explanations above, we can conclude that the three bounded functions are:
1. The Reciprocal Function
2. The Square Root Function
3. The Absolute Value Function
Now, let's determine which three of the twelve basic functions are even. A function is said to be even if it satisfies the property f(x) = f(-x) for all values of x.
The three even functions among the twelve basic functions are:
1. The Identity Function (f(x) = x) - It is symmetric with respect to the y-axis.
2. The Squaring Function (f(x) = x^2) - It is symmetric with respect to the y-axis.
3. The Cosine Function (f(x) = cos(x)) - It is symmetric with respect to the y-axis.
Therefore, the three even functions are the Identity Function, the Squaring Function, and the Cosine Function.