A quadratic function has the form ax2 + bx + c: f(x) = 2x2 + 3x + 4 Examples of functions with fractions include: f(x) = (1/x), f(x) = (x + 1)/(x - 1), etc. Functions with a root include: f(x) = √x, f(x) = √(x2 + 1), f(x) = √-x, etc.

For example, a domain of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10. Always use parentheses if you are a using the infinity symbol, ∞. This is because infinity is a concept and not a number.

To get an idea of the function choose any x-value and plug it into the function. Solving the function with this x-value will output a y-value. These x- and y-values are a coordinate (x, y) of the graph of the function. Plot this coordinate and repeat the process with another x-value. Plotting a few values in this fashion should give you a general idea of shape of the quadratic function.

To get an idea of the function choose any x-value and plug it into the function. Solving the function with this x-value will output a y-value. These x- and y-values are a coordinate (x, y) of the graph of the function. Plot this coordinate and repeat the process with another x-value. Plotting a few values in this fashion should give you a general idea of shape of the quadratic function.

For example: Identify the domain of the function f(x) = (x + 1)/(x - 1). The denominator of this function is (x - 1). Set it equal to zero and solve for x: x – 1 = 0, x = 1. Write the domain: The domain of this function cannot include 1, but includes all real numbers except 1; therefore, the domain is (-∞, 1) U (1, ∞). (-∞, 1) U (1, ∞) can be read as the set of all real numbers excluding 1. The infinity symbol, ∞, represents all real numbers. In this case, all real numbers greater than 1 and less than one are included in the domain.

For example: Identify the domain of the function f(x) = √(x + 3). The terms within the radical are (x + 3). Set them greater than or equal to zero: (x + 3) ≥ 0. Solve for x: x ≥ -3. The domain of this function includes all real numbers greater than or equal to -3; therefore, the domain is [-3, ∞).

The easiest way to identify the range of other functions, such as root and fraction functions, is to draw the graph of the function using a graphing calculator.

For example, find the range of 3x2 + 6x -2. Calculate x-coordinate of vertex: x = -b/2a = -6/(2*3) = -1

Calculate y-coordinate: y = 3x2 + 6x – 2 = 3(-1)2 + 6(-1) -2 = -5. The vertex of this function is (-1, -5).

Use the x-value -2: y = 3x2 + 6x – 2 = y = 3(-2)2 + 6(-2) – 2 = 12 -12 -2 = -2. This yields the coordinate (-2, -2). This coordinate tells you that the parabola continues above the vertex (-1, -5); therefore, the range encompasses all y-values above -5. The range of this function is [-5, ∞)

For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10. Always use parentheses if you are a using the infinity symbol, ∞.

Some root functions will start above or below the x-axis. In this case, the range is determined by the point the root function starts. If the parabola starts at y = -4 and goes up, then the range is [-4, +∞). The easiest way to graph a function is to use a graphing program or a graphing calculator. If you do not have a graphing calculator, you can draw a rough sketch of a graph by plugging x-values into the function and getting the corresponding y-values. Plot these coordinates on the graph to get an idea of the shape of the graph.

A fraction function will include all points except those at the asymptote. They often have ranges such as (-∞, 6) U (6, ∞).

For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10. Always use parentheses if you are a using the infinity symbol, ∞.