To find the domain and range of a function, follow these steps: How to Find the Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- Check if the function can include all real numbers or if there are restrictions. For example, values that make the denominator zero or result in a negative number under a square root are excluded.
- The domain is all real numbers except the values where the function is undefined.
- Sometimes domain limits are given explicitly as intervals.
How to Find the Range:
- The range is the set of all possible output values (y-values) the function can produce.
- Write the function as y = f(x), then solve for x to get x = g(y).
- The domain of the function x = g(y) corresponds to the range of the original function y = f(x).
- Alternatively, imagine or graph the function to see what y-values it takes.
- For rational functions, solve for x in terms of y and exclude values that cause undefined x.
Example:
For f(x)=1x2−1f(x)=\frac{1}{x^2-1}f(x)=x2−11,
- The function is undefined for x=±1x=\pm 1x=±1 (denominator zero), so
- Domain: All real numbers except ±1\pm 1±1.
For f(x)=x2+1f(x)=x^2+1f(x)=x2+1,
- Domain: All real numbers (no restrictions).
- Range: Since x2≥0x^2\geq 0x2≥0, the smallest value is 1, so range is [1,∞)[1,\infty)[1,∞).
General Summary:
- Domain: All valid inputs xxx where the function is defined.
- Range: All possible outputs yyy of the function.
These steps work for various function types including polynomial, rational, exponential, and square root functions. Specific domain and range depend on the function's formula and restrictions.
