Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=8x+12, g(x)=3x-1

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation $(g\circ f)(x)$ means $g(f(x))$, which involves substituting $f(x)$ into $g(x)$.
2. Step 2: Substitute $f(x)=8x+12$ into $g(x)=3x-1$. This means replacing every $x$ in $g(x)$ with $8x+12$.
3. Step 3: Write the expression for $g(f(x))$ as $g(8x+12)=3(8x+12)-1$.
4. Step 4: Simplify the expression $3(8x+12)-1$ by distributing the 3 and then combining like terms.
5. Step 5: Determine the domain of $(g\circ f)(x)$. Since both $f(x)$ and $g(x)$ are linear functions, their domains are all real numbers. Therefore, the domain of $(g\circ f)(x)$ is also all real numbers.

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 we first apply the function f to x, and then apply the function g to the result of f. Understanding how to correctly perform this operation is essential for solving the problem.

Finding the Domain

The domain of a function is the set of all possible input values (x) for which the function is defined. When composing functions, the domain of the composite function (g∘ƒ)(x) is determined by the domain of f and the values that g can accept based on the outputs of f. Analyzing these domains ensures that we only consider valid inputs.

Linear Functions

Both functions f(x) = 8x + 12 and g(x) = 3x - 1 are linear functions, characterized by their constant rate of change and graphing as straight lines. Understanding the properties of linear functions, such as their slopes and intercepts, is crucial for evaluating their compositions and determining the overall behavior of (g∘ƒ)(x).