In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = 3/(x-4) and g(x) = 3/x + 4

Verified step by step guidance

1. To find $f(g(x))$, substitute $g(x)$ into $f(x)$. Start with $f(x)=\frac{3}{x-4}$ and $g(x)=\frac{3}{x}+4$.
2. Replace $x$ in $f(x)$ with $g(x)$: $f(g(x))=\frac{3}{(\frac{3}{x}+4)-4}$. Simplify the expression inside the denominator.
3. To find $g(f(x))$, substitute $f(x)$ into $g(x)$. Start with $g(x)=\frac{3}{x}+4$ and $f(x)=\frac{3}{x-4}$.
4. Replace $x$ in $g(x)$ with $f(x)$: $g(f(x))=\frac{3}{\frac{3}{x-4}}+4$. Simplify the expression.
5. To determine if $f$ and $g$ are inverses, check if $f(g(x))=x$ and $g(f(x))=x$. If both are true, then $f$ and $g$ are inverses.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For functions f and g, the composition f(g(x)) means applying g first and then f to the result. This process is essential for evaluating how two functions interact and can reveal properties such as inverses.

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f and g are inverses, then f(g(x)) = x and g(f(x)) = x for all x in their domains. Identifying whether two functions are inverses is crucial in understanding their relationship and can be determined through function composition.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. When analyzing functions and their inverses, it is important to consider the domains and ranges to ensure that the compositions f(g(x)) and g(f(x)) are valid. This understanding helps in determining the behavior and limitations of the functions involved.