Given functions f and g, find (b)$(g\circ \text{ƒ})\left(x\right)$ and its domain. See Examples 6 and 7.
$\text{ƒ}\left(x\right)=8x+12,g\left(x\right)=3x-1$

Verified step by step guidance

1. Recall that the composition of functions \( (g \circ f)(x) \) means you substitute \( f(x) \) into \( g(x) \). In other words, \( (g \circ f)(x) = g(f(x)) \).
2. Given \( f(x) = 8x + 12 \) and \( g(x) = 3x - 1 \), substitute \( f(x) \) into \( g \) to get \( g(f(x)) = 3(8x + 12) - 1 \).
3. Simplify the expression by distributing the 3 across \( 8x + 12 \), which gives \( 3 \times 8x + 3 \times 12 - 1 \).
4. Combine like terms to write the simplified form of \( (g \circ f)(x) \).
5. To find the domain of \( (g \circ f)(x) \), consider the domain of \( f(x) \) first, then ensure that the output of \( f(x) \) fits within 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 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(x), then use that output as the input for g. Understanding this process is essential to correctly find (g∘f)(x).

Evaluating Linear Functions

Both f(x) = 8x + 12 and g(x) = 3x - 1 are linear functions. Evaluating these functions means substituting a value for x and simplifying. Knowing how to substitute and simplify expressions is key to finding the composite function.

Domain of Composite Functions

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 both f and g are linear with domains of all real numbers, the composite function's domain is also all real numbers, but this concept is crucial for more complex functions.