how to find range of a function

To find the range of a function, which is the set of all possible output values (y-values), you can use several methods depending on how the function is given (formula, graph, or relation). Here are the key approaches:

1. Algebraic Method (Using the Formula)

• Start with the function written as y=f(x)y=f(x)y=f(x).
• Solve the equation for xxx in terms of yyy (i.e., find the inverse relation if possible).
• Determine for which yyy-values the inverse is defined (exclude values that make the function undefined).
• The set of these yyy-values is the range of the original function

2. Graphical Method

• Look at the graph of the function.
• Identify all the yyy-values that the graph attains from bottom to top.
• If the graph is continuous, the range is all yyy-values covered by the graph.
• Exclude any gaps or holes where the function is not defined

3. Using a Relation (Set of Ordered Pairs)

• Write down the set of ordered pairs (x,y)(x,y)(x,y).
• Extract the yyy-coordinates from these pairs.
• The set of these yyy-values is the range

Additional Tips and Examples

• For common functions, there are known range rules:
  • Linear functions: range is all real numbers.
  • Quadratic functions y=a(x−h)2+ky=a(x-h)^2+ky=a(x−h)2+k: range is y≥ky\ge ky≥k if a>0a>0a>0, or y≤ky\le ky≤k if a<0a<0a<0.
  • Square root functions: range is y≥0y\ge 0y≥0.
  • Exponential functions: range is y>0y>0y>0.
  • Logarithmic functions: range is all real numbers

• Another efficient method is to find the inverse function and then determine its domain; this domain corresponds to the range of the original function

Summary of Steps to Find Range Algebraically:

1. Write the function as y=f(x)y=f(x)y=f(x).
2. Swap xxx and yyy and solve for yyy (find inverse).
3. Identify the domain of the inverse function.
4. The domain of the inverse is the range of the original function.
5. Write the range accordingly, excluding values that make the function undefined

This approach works well especially when the inverse can be found explicitly and helps to handle more complicated functions where direct inspection is difficult.