Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5.
$(g\circ g)(-2)$

Verified step by step guidance

1. Understand that the notation \((g \circ g)(-2)\) means you need to find \(g(g(-2))\), which is the composition of the function \(g\) with itself, evaluated at \(-2\).
2. First, find the inner function value \(g(-2)\) by substituting \(-2\) into the function \(g(x) = -x + 3\). This means calculating \(g(-2) = -(-2) + 3\).
3. Simplify the expression from the previous step to get the value of \(g(-2)\).
4. Next, take the result from step 3 and substitute it back into the function \(g(x)\) to find \(g(g(-2))\). This means calculating \(g(\text{result from step 3}) = -\text{result} + 3\).
5. Simplify the expression from step 4 to find the final value of \((g \circ g)(-2)\).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘g)(x) = g(g(x)). It requires evaluating the inner function first, then using that output as the input for the outer function.

Evaluating Functions at a Given Input

Evaluating a function at a specific input means substituting the input value into the function's formula and simplifying to find the output. For example, g(-2) means replacing x with -2 in g(x).

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are straightforward to evaluate and compose, as seen in the given functions f(x) = 2x - 3 and g(x) = -x + 3.