In Exercises 91–100, find all values of x satisfying the given conditions. y = (x - 5)^(3/2) and y = 125

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. In this context, the function y = (x - 5)^(3/2) represents a transformation of the basic power function, where the exponent affects the shape and behavior of the graph. Understanding how to manipulate and solve equations involving exponents is crucial for finding the values of x that satisfy the given conditions.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In this case, we need to set the two equations equal to each other: (x - 5)^(3/2) = 125. This requires isolating x, which may involve taking roots or applying inverse operations. Mastery of algebraic techniques is essential for effectively solving such equations.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values) that the function can accept, while the range refers to the set of possible output values (y-values). For the function y = (x - 5)^(3/2), the domain is restricted to x ≥ 5, as negative values under the square root are not defined in the real number system. Understanding the domain is important for determining valid solutions to the equation.