Let $\text{ƒ}\left(x\right)=x^2+3$ and $g\left(x\right)=-2x+6$. Find each of the following. See Example 1.
$\left(\text{ƒ}g\right)(-3)$

Verified step by step guidance

1. Understand that (ƒg)(x) means the composition of functions ƒ and g, which is ƒ(g(x)). This means you first apply g to x, then apply ƒ to the result.
2. Start by finding g(-3) by substituting -3 into the function g(x) = -2x + 6. Write the expression: \(g(-3) = -2(-3) + 6\).
3. Simplify the expression for g(-3) to find the value that will be input into ƒ. This involves multiplying and adding the numbers inside the parentheses.
4. Next, substitute the value found for g(-3) into the function ƒ(x) = x^2 + 3. Write the expression: \(ƒ(g(-3)) = (g(-3))^2 + 3\).
5. Finally, simplify the expression by squaring the value of g(-3) and then adding 3 to find the value of (ƒg)(-3).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (ƒg)(x) = ƒ(g(x)). It means you first evaluate g at x, then use that output as the input for ƒ. This concept is essential for understanding how to combine functions and evaluate them at specific values.

Evaluating Functions at a Given Input

Evaluating a function at a given input means substituting the input value into the function's formula and simplifying to find the output. For example, to find g(-3), replace x with -3 in g(x) = -2x + 6 and simplify. This step is crucial before composing functions.

Polynomial and Linear Functions

ƒ(x) = x² + 3 is a polynomial function, and g(x) = -2x + 6 is a linear function. Understanding their forms helps in correctly substituting and simplifying expressions during evaluation and composition. Recognizing these types aids in anticipating the behavior and complexity of the functions.