1. The problem is to find the domain and range of a combined function, which typically means a function formed by operations like addition, subtraction, multiplication, division, or composition of two functions. 2. The domain of a combined function is the set of all input values for which the combined function is defined. This usually involves the intersection of the domains of the individual functions and considering any restrictions such as division by zero or square roots of negative numbers. 3. The range of a combined function is the set of all possible output values it can produce. 4. To find the domain: - Identify the domains of the individual functions. - Find the intersection of these domains. - Exclude any values that cause undefined expressions (e.g., division by zero). 5. To find the range: - Analyze the behavior of the combined function. - Consider the ranges of the individual functions and how the combination affects possible outputs. 6. Without specific functions given, the general approach is: $$\text{Domain}(f \circ g) = \{x \mid x \in \text{Domain}(g) \text{ and } g(x) \in \text{Domain}(f)\}$$ 7. For example, if $f(x) = \sqrt{x}$ and $g(x) = x - 1$, then: - Domain of $g$ is all real numbers. - Domain of $f$ is $x \geq 0$. - So domain of $f \circ g$ is $x$ such that $g(x) \geq 0$, i.e., $x - 1 \geq 0 \Rightarrow x \geq 1$. 8. The range depends on the output of $f(g(x))$ for $x$ in the domain. Since no specific functions are provided, this is the general method to find domain and range of combined functions.