Given functions f and g, find (a)(ƒ∘g)(x) and its domain. See Examples 6 and 7. ƒ(x)=-6x+9, g(x)=5x+7

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 the function $g(x)$ into the function $f(x)$.
2. Step 2: Substitute $g(x)=5x+7$ into $f(x)=-6x+9$. This means replacing every $x$ in $f(x)$ with $5x+7$.
3. Step 3: Write the expression for $f(g(x))$ as $-6(5x+7)+9$.
4. Step 4: Simplify the expression $-6(5x+7)+9$ by distributing $-6$ and combining like terms.
5. Step 5: Determine the domain of $(f\circ g)(x)$. Since both $f(x)$ and $g(x)$ are linear functions, their domains are all real numbers. Therefore, the domain of $(f\circ g)(x)$ 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, (ƒ∘g)(x) means applying g first and then applying f to the result of g. This process is essential for evaluating the composite 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 finding the domain of a composite function, it is crucial to consider the domains of both individual functions and any restrictions that may arise from the composition process, such as division by zero or square roots of negative numbers.

Linear Functions

Linear functions are polynomial functions of degree one, represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this question, both f(x) and g(x) are linear functions, which means their graphs are straight lines. Understanding their properties helps in analyzing their composition and determining the resulting function's behavior.