Which of the following graphs have the same domain? This question is often encountered in mathematics, particularly when analyzing functions and their graphs. The domain of a function refers to the set of all possible input values for which the function is defined. Understanding the domain is crucial for determining the behavior and properties of a function. In this article, we will explore various graphs and identify those that share the same domain.
The domain of a function can be influenced by several factors, such as the presence of square roots, denominators, or absolute values. By examining the graphs of different functions, we can determine which ones have the same domain.
Consider the following examples:
1. Graph A: f(x) = √(x^2 – 4)
Graph B: g(x) = √(x^2 – 9)
Both Graph A and Graph B are defined for all real numbers except for the values that make the expressions inside the square roots negative. In this case, both functions are defined for x ≥ 2 and x ≤ -2. Therefore, Graph A and Graph B have the same domain.
2. Graph C: h(x) = 1/(x – 3)
Graph D: k(x) = 1/(x + 2)
Graph C and Graph D are rational functions, and their domains are determined by the values that make the denominators equal to zero. In this case, both functions are defined for all real numbers except for x = 3 and x = -2, respectively. Hence, Graph C and Graph D have the same domain.
3. Graph E: m(x) = |x – 5|
Graph F: n(x) = √(x^2 – 25)
Graph E represents an absolute value function, and its domain is all real numbers since the absolute value of any real number is defined. On the other hand, Graph F is defined for all real numbers except for x = 5 and x = -5, as the expression inside the square root must be non-negative. Therefore, Graph E and Graph F do not have the same domain.
In conclusion, identifying which graphs have the same domain requires analyzing the functions and their expressions. By considering factors such as square roots, denominators, and absolute values, we can determine the domain of each function and identify those that share the same domain. This understanding is essential for further analysis and manipulation of functions in various mathematical contexts.
