# 60 2 1 Practice Power And Radical Functions

## Understanding Power Functions

### What are Power Functions?

Power functions are mathematical expressions that involve raising a variable to a constant power. They have the general form:

f(x) = axn

Here, 'a' represents the coefficient, 'x' is the variable, and 'n' is the exponent. Power functions can have positive or negative exponents.

### Properties of Power Functions

Power functions exhibit several important properties:

1. When 'n' is a positive integer, the graph of the function passes through the origin (0, 0) and increases or decreases rapidly depending on the value of 'n'.
2. When 'n' is an even positive integer, the graph of the function is symmetric with respect to the y-axis.
3. When 'n' is an odd positive integer, the graph of the function is not symmetric.
4. When 'n' is a negative integer, the graph of the function is a reciprocal of the corresponding positive power function.

### Examples of Power Functions

Let's look at a few examples of power functions:

1. f(x) = 3x2
2. f(x) = -2x3
3. f(x) = 4x-1

Each of these examples represents a different type of power function, showcasing the various possibilities within this category of functions.

Radical functions involve the use of radicals, which are mathematical symbols indicating the root of a number. They have the general form:

f(x) = a√(x - h) + k

Here, 'a' represents the coefficient, 'x' is the variable, 'h' is the horizontal shift, and 'k' is the vertical shift. The radical function involves taking the square root, cube root, or nth root of the expression inside the radical symbol.

Radical functions exhibit several important properties:

1. The domain of a radical function is restricted to values that make the expression inside the radical non-negative.
2. The graph of a radical function may have vertical and/or horizontal shifts.
3. The graph of a radical function may have restrictions on the range.

Let's look at a few examples of radical functions:

1. f(x) = √(x + 3)
2. f(x) = 2√(x - 2) + 1
3. f(x) = 3√(x2 - 4)

Each of these examples showcases a different type of radical function, demonstrating the versatility and applicability of this category of functions.

## Practicing Power and Radical Functions

### Power Functions Practice

Now that we have a good understanding of power functions, let's practice solving some problems:

1. Solve the equation f(x) = 2x3 - 5x2 + 3x - 1 for x = 2.
2. Find the value of 'a' in the power function f(x) = ax4 - 4x3 + 6x2 - 4x + 1 if the graph passes through the point (1, 2).
3. Graph the power function f(x) = 2x-2 and determine its domain and range.