Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=2/x, g(x)=x+1

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation $(f\circ g)(x)$ means $f(g(x))$, which involves substituting $g(x)$ into $f(x)$.
2. Step 2: Identify the given functions. Here, $f(x)=\frac{2}{x}$ and $g(x)=x+1$.
3. Step 3: Substitute $g(x)$ into $f(x)$. This means replacing every $x$ in $f(x)$ with $g(x)$, resulting in $f(g(x))=\frac{2}{x+1}$.
4. Step 4: Determine the domain of $(f\circ g)(x)$. The domain of a function is the set of all possible input values ($x$) that do not lead to division by zero or other undefined operations.
5. Step 5: For $f(g(x))=\frac{2}{x+1}$, the expression is undefined when $x+1=0$. Solve $x+1=0$ to find $x=-1$. Therefore, the domain of $(f\circ g)(x)$ is all real numbers except $x=-1$.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, (ƒ∘g)(x) means applying g first, then applying f to the result of g. This process is essential for evaluating the combined function and understanding how the two functions interact.

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 domains of the individual functions and any restrictions that arise from their composition, such as division by zero.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, f(x) = 2/x is a rational function, which is undefined when the denominator is zero. Understanding the properties of rational functions is crucial for determining their domains and identifying any restrictions that may affect the composition.