Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5.
$open paren g composed with g close paren of negative 2$

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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 $-2$.

First, find the inner function value $g(-2)$ by substituting $-2$ into the function $g(x) = -x + 3$. This means calculating $g(-2) = -(-2) + 3$.

Simplify the expression from the previous step to get the value of $g(-2)$.

Next, take the result from step 3 and substitute it back into the function $g(x)$ to find $g(g(-2))$. This means calculating $g(\text{result from step 3}) = -\text{result} + 3$.

Simplify the expression from step 4 to find the final value of $(g \circ g)(-2)$.

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주요 개념

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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.

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