A power function, or exponential function, is a mathematical function that relates a variable to a constant raised to the power of that variable. The constant is known as the base, and the variable is known as the exponent. Power functions have many applications, including modeling exponential growth and decay, as well as solving equations. They are also used in physics, engineering, and economics.
Power Functions
A power function is a function of the form f(x) = x^n, where n is a real number. The graph of a power function is a curve that passes through the origin. The shape of the graph depends on the value of n.
- If n is a positive integer, the graph of the power function is a parabola that opens up.
- If n is a negative integer, the graph of the power function is a parabola that opens down.
- If n is a rational number that is not an integer, the graph of the power function is a curve that has a cusp at the origin.
- If n is an irrational number, the graph of the power function is a smooth curve.
The following table shows the graphs of some power functions:
| n | Graph |
|---|---|
| 2 | Parabola that opens up |
| -2 | Parabola that opens down |
| 1/2 | Curve that has a cusp at the origin |
| π | Smooth curve |
Power functions have a number of important properties.
- The domain of a power function is the set of all real numbers.
- The range of a power function is the set of all positive real numbers if n is even, and the set of all negative real numbers if n is odd.
- The graph of a power function is symmetric about the y-axis if n is even, and symmetric about the origin if n is odd.
- The derivative of a power function is f'(x) = nx^(n-1).
- The integral of a power function is f(x) = (1/(n+1))x^(n+1) + C, where C is a constant of integration.
Power functions are used in a variety of applications, including physics, engineering, and economics. For example, the force of gravity between two objects is a power function of the distance between the objects. The power law of economics states that the demand for a good is a power function of its price.
Question 1:
What is the definition of a power function?
Answer:
A power function is a mathematical function that has a variable raised to a constant exponent. The variable is the base, and the exponent is the power.
Question 2:
What are the properties of a power function?
Answer:
Power functions have the following properties:
- The graph of a power function is a curve that is either increasing or decreasing.
- The rate of change of a power function is proportional to the power of the variable.
- Power functions are often used to model exponential growth or decay.
Question 3:
How can you differentiate a power function?
Answer:
The derivative of a power function is equal to the power of the variable multiplied by the original function. The formula for the derivative of a power function is:
f(x) = x^n
f'(x) = nx^(n-1)
Well, there you have it, folks! Now you know the ins and outs of power functions. They might seem a bit intimidating at first, but trust me, they’re not as scary as they look. Just remember the basics, practice a bit, and you’ll be a power function pro in no time. Thanks for sticking with me on this journey into the world of math. Be sure to swing by again soon for more mathematical adventures!