To find the domain and range of a function, you need to consider the inputs (domain) and outputs (range) that make the function meaningful and defined.

Domain (Input Values)

The domain of a function is the set of all possible input values (usually $x$) for which the function is defined. To find the domain, you should consider the following common restrictions:

1. Denominators: The function is undefined when the denominator is zero. Solve for when the denominator equals zero and exclude those values from the domain. $\text{Example: } f(x) = \frac{1}{x-2} \quad \text{Domain: } x \neq 2$
2. Square roots: For square roots (or even roots), the expression inside the root must be non-negative (i.e., $\geq 0$). $\text{Example: } f(x) = \sqrt{x-3} \quad \text{Domain: } x \geq 3$
3. Logarithms: The argument of a logarithmic function must be positive. $\text{Example: } f(x) = \log(x-1) \quad \text{Domain: } x > 1$
4. Other functions: Consider any other constraints specific to the function (e.g., trig functions like $\tan(x)$ are undefined at certain points).

Range (Output Values)

The range is the set of all possible output values (usually $y$) the function can take. To find the range:

1. Identify the function behavior: Look at how the function behaves as $x$ approaches large values, small values, or restricted points (from the domain).
2. Check boundaries: If the function has a restricted domain (e.g., a square root function), check the possible output values by analyzing the minimum and maximum outputs. $\text{Example: } f(x) = \sqrt{x} \quad \text{Range: } y \geq 0$
3. Inverse function approach: Sometimes, finding the inverse of the function and determining the domain of the inverse helps to find the range of the original function.

Examples:

1. Linear Function $f(x) = 2x + 3$:

   • Domain: All real numbers, $(-\infty, \infty)$
   • Range: All real numbers, $(-\infty, \infty)$

2. Square Root Function $f(x) = \sqrt{x - 1}$:

   • Domain: $x \geq 1$
   • Range: $y \geq 0$

3. Rational Function $f(x) = \frac{1}{x-4}$:

   • Domain: $x \neq 4$
   • Range: $y \neq 0$ (since the function never equals 0)

Would you like a specific example to work through or more detailed explanations on certain types of functions?

5 Related Questions:

1. How do you find the domain and range of composite functions?
2. What are the restrictions for logarithmic and exponential functions?
3. How does finding the domain differ for implicit functions?
4. How do asymptotes affect the range of rational functions?
5. How do transformations (shifts, stretches) affect the domain and range?

Tip:

For complex functions, try breaking them down into simpler components (like identifying domains for each part) before combining the results.