To find the domain and range of a function, follow these steps:
Domain of a Function
The domain is the set of all possible input values (x-values) for which the function is defined. How to find the domain:
- Identify values of xxx that make the function undefined.
- Common restrictions include:
- Division by zero (denominator ≠ 0).
- Even roots (like square roots) require the radicand to be ≥ 0.
- Logarithmic functions require the argument > 0.
- For polynomials, the domain is all real numbers since there are no restrictions.
- For rational functions, solve for values that make the denominator zero and exclude them.
- For square root functions, solve inequalities to ensure the radicand is non-negative.
Example:
For f(x)=x+3f(x)=\sqrt{x+3}f(x)=x+3, set x+3≥0x+3\geq 0x+3≥0 which gives
x≥−3x\geq -3x≥−3. So, the domain is [−3,∞)[-3,\infty)[−3,∞)
Range of a Function
The range is the set of all possible output values (y-values) the function can produce. How to find the range:
- Consider the values y=f(x)y=f(x)y=f(x) can take when xxx is in the domain.
- Analyze the function behavior or graph it.
- For example, the range of f(x)=∣ax+b∣f(x)=|ax+b|f(x)=∣ax+b∣ is all non-negative real numbers [0,∞)[0,\infty)[0,∞) because absolute values are always ≥ 0.
- For f(x)=x−1f(x)=\sqrt{x-1}f(x)=x−1, the output is always ≥ 0, so range is [0,∞)[0,\infty)[0,∞).
Example:
If 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,∞)
Summary of Steps
- Domain: Find all xxx values where the function is defined by excluding values that cause division by zero or negative radicands.
- Range: Determine all possible yyy values produced by the function over its domain, often by analyzing the function or its graph.
This approach applies to various types of functions including polynomials, rational, radical, logarithmic, and absolute value functions
