Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=√x, g(x)=1/(x+5)

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (g∘ƒ)(x) means applying function f first, followed by function g. Understanding how to correctly substitute and evaluate these functions is crucial for finding the composed function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When composing functions, the domain of the resulting function is determined by the restrictions imposed by both functions involved. It is essential to identify any values that would make the functions undefined, such as division by zero or taking the square root of a negative number.

Square Root and Rational Functions

Square root functions, like ƒ(x)=√x, are only defined for non-negative inputs (x ≥ 0), while rational functions, such as g(x)=1/(x+5), are undefined when the denominator equals zero (x ≠ -5). Understanding these properties helps in determining the overall domain of the composed function, ensuring that all input values are valid for both functions.