To find the domain and range of a function, follow these steps:

Finding the Domain

The domain is the set of all possible input values (usually xxx) for which the function is defined.

• Identify restrictions on xxx:
  • For functions with denominators, set the denominator ≠0\neq 0=0 and solve for xxx to exclude values that make the denominator zero.
  • For functions with even roots (like square roots), set the radicand ≥0\geq 0≥0 and solve for xxx to ensure the expression under the root is non-negative.
  • For other functions like polynomials, the domain is usually all real numbers since there are no restrictions.
• Express the domain in interval notation , excluding any restricted values.

Example: For f(x)=2x−3f(x)=\frac{2}{x-3}f(x)=x−32​, set x−3≠0x-3\neq 0x−3=0, so x≠3x\neq 3x=3. The domain is all real numbers except 3, or (−∞,3)∪(3,∞)(-\infty,3)\cup (3,\infty)(−∞,3)∪(3,∞)

Finding the Range

The range is the set of all possible output values (usually yyy or f(x)f(x)f(x)) the function can produce.

• Analyze the function or its graph to determine the possible yyy-values.
• For some functions, the range can be deduced from the domain and the function's behavior:
  • For example, a square root function f(x)=x−1f(x)=\sqrt{x-1}f(x)=x−1​ has range [0,∞)[0,\infty)[0,∞) because square roots are never negative.
  • For quadratic functions like f(x)=x2+2f(x)=x^2+2f(x)=x2+2, since x2≥0x^2\geq 0x2≥0, the minimum value of f(x)f(x)f(x) is 2, so the range is [2,∞)[2,\infty)[2,∞).
• Use the graph to see the lowest and highest points or asymptotes to help determine the range.

Example: For f(x)=x−1f(x)=\sqrt{x-1}f(x)=x−1​, domain is [1,∞)[1,\infty)[1,∞) and range is [0,∞)[0,\infty)[0,∞)

Summary

• Domain: Find all valid xxx-values by excluding values that cause division by zero or negative radicands.
• Range: Find all possible yyy-values by analyzing the function's outputs or graph.

This approach applies to various types of functions including polynomials, rational functions, square root functions, absolute value functions, and exponential functions