Understanding Domain and Range of Functions
When we talk about functions in math, two important ideas are domain and range. These concepts change depending on the type of function we're looking at.
Let's break it down by function types:
-
Linear Functions:
- Shape: These follow the formula ( f(x) = mx + b ). Here, ( m ) and ( b ) are numbers.
- Domain: You can use any number for ( x ). So, the domain is all real numbers, written as ( (-\infty, \infty) ).
- Range: Just like the domain, the range is also all real numbers, ( (-\infty, \infty) ).
-
Quadratic Functions:
- Shape: These follow the formula ( f(x) = ax^2 + bx + c ). Remember, ( a ) can't be zero.
- Domain: You can also use any number for ( x ), so the domain is all real numbers, ( (-\infty, \infty) ).
- Range: This one depends on the value of ( a ):
- If ( a ) is more than zero, the range is from a certain point ( k ) going up (written as ( [k, \infty) )).
- If ( a ) is less than zero, the range goes down from ( k ) (written as ( (-\infty, k] )). Here, ( k ) is the highest point on the graph.
-
Rational Functions:
- Shape: These look like ( f(x) = \frac{p(x)}{q(x)} ), but ( q(x) ) can't be zero.
- Domain: You can use all real numbers except where ( q(x) ) is zero.
- Range: This can change a lot. Some values might not be included.
-
Exponential Functions:
- Shape: These follow the formula ( f(x) = a \cdot b^x ), where ( b ) is a positive number.
- Domain: Any number can be used, so the domain is all real numbers, ( (-\infty, \infty) ).
- Range: The output is from 0 to infinity, written as ( (0, \infty) ).
-
Logarithmic Functions:
- Shape: These follow the formula ( f(x) = \log_b(x) ), where ( b ) is a positive number.
- Domain: You can only use positive numbers, so the domain is ( (0, \infty) ).
- Range: Just like linear functions, the range is all real numbers, ( (-\infty, \infty) ).
By understanding domain and range for different types of functions, we can get a better idea of what their graphs look like and how they behave.