Final answer:
The domain of the function f(x) = √(x² + 3x - 7) is x ≤ -7 or x ≥ 1. The function does not exhibit any symmetry.
Step-by-step explanation:
The domain of a function represents the set of all possible input values that the function can accept. In this case, the function is f(x) = √(x² + 3x - 7). To find the domain, we need to ensure that the expression inside the square root is non-negative, since the square root of a negative number is undefined. Setting x² + 3x - 7 ≥ 0, we can now solve for x to determine the domain.
By factoring the quadratic expression, we get (x - 1)(x + 7) ≥ 0. This means that either both factors are positive or both factors are negative. We can use a sign chart or test values to find the solution set. The domain is:
x ≤ -7 or x ≥ 1.
The function f(x) = √(x² + 3x - 7) does not exhibit any symmetry, as it is neither even nor odd. Even functions are symmetric about the y-axis, meaning that if (x, y) is a point on the graph, then (-x, y) is also a point. Odd functions, on the other hand, are symmetric about the origin, meaning that if (x, y) is a point on the graph, then (-x, -y) is also a point.