Given functions f and g, (b)$(g\circ \text{ƒ})\left(x\right)$ and its domain. See Examples 6 and 7.
$\text{ƒ}\left(x\right)=-6x+9,g\left(x\right)=5x+7$

Verified step by step guidance

1. Recall that the composition of functions \((g \circ f)(x)\) means \(g(f(x))\), which is the function \(g\) evaluated at \(f(x)\).
2. Start by substituting \(f(x)\) into \(g(x)\). Since \(f(x) = -6x + 9\), replace the \(x\) in \(g(x) = 5x + 7\) with \(-6x + 9\). This gives \(g(f(x)) = 5(-6x + 9) + 7\).
3. Simplify the expression by distributing the 5 and then combining like terms: \(5(-6x + 9) + 7 = 5 \times (-6x) + 5 \times 9 + 7\).
4. Write the simplified form of \((g \circ f)(x)\) after performing the multiplication and addition.
5. Determine the domain of \((g \circ f)(x)\) by considering the domain of \(f(x)\) and the domain of \(g(x)\). Since both \(f\) and \(g\) are linear functions, their domains are all real numbers, so the domain of \((g \circ f)(x)\) is also all real numbers.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘f)(x) = g(f(x)). It means you first evaluate f at x, then use that output as the input for g. Understanding this process is essential to correctly combine the given functions.

Domain of a Composite Function

The domain of (g∘f)(x) consists of all x-values in the domain of f for which f(x) is in the domain of g. Since g is applied after f, you must ensure that the output of f lies within g's domain to find the composite function's valid inputs.

Linear Functions and Their Domains

Both f(x) = -6x + 9 and g(x) = 5x + 7 are linear functions, which have domains of all real numbers. This simplifies finding the composite function's domain, as there are no restrictions from either function's domain or range.