How to find domain and range algebraically?
Understand the Problem
The question is asking how to determine the domain and range of a function using algebraic methods. This typically involves identifying the possible input values (domain) and the resulting output values (range) from the function's expression.
Answer
Determine the domain by identifying restrictions. Determine the range by analyzing the function's behavior.
Answer for screen readers
To find the domain and range of a function algebraically, identify any restrictions on the variable to find the domain and analyze the behavior of the function for the range.
Steps to Solve
- Determine the domain of the function
To find the domain, identify any values of the variable that would make the function undefined. For example:
- For a rational function (with a denominator), set the denominator not equal to zero and solve for the variable.
- For a square root function, set the expression inside the root greater than or equal to zero and solve for the variable.
- Solve inequalities or equations for domain
Solve the inequalities or equations obtained in the first step to determine the set of all possible input values. For example: $$ f(x) = \frac{1}{x} \implies x \ne 0 $$ $$ g(x) = \sqrt{x - 2} \implies x - 2 \geq 0 \implies x \geq 2 $$
- Determine the range of the function
To find the range, first express the function in terms of the dependent variable and then solve for the independent variable if possible. Study the behavior of the function (increasing, decreasing, asymptotes, etc.) to determine the set of all possible output values.
- Analyze function behavior for range
Analyze the function's behavior by considering limits and any restrictions on the dependent variable. For example, if the function is $ f(x) = x^2 + 1 $, notice that the minimum value is 1 (since $ x^2 \geq 0 $), so the range is $ [1, \infty) $.
To find the domain and range of a function algebraically, identify any restrictions on the variable to find the domain and analyze the behavior of the function for the range.
More Information
When determining domains, common restrictions come from denominators and roots. For ranges, analyzing the function's overall behavior helps identify all possible outputs.
Tips
Common mistakes include forgetting to consider all restrictions on the domain (like negative values under a square root) and not thoroughly analyzing the function's behavior for all possible outputs.
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