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

Understand that the notation $(ƒ \circ ƒ)(2)$ means you need to find $ƒ(ƒ(2))$, which is the composition of the function $ƒ$ with itself, evaluated at $x=2$.

First, find the inner function value $ƒ(2)$ by substituting $x=2$ into the function $ƒ(x) = 2x - 3$. This gives $ƒ(2) = 2(2) - 3$.

Next, take the result from $ƒ(2)$ and substitute it back into the function $ƒ(x)$ to find $ƒ(ƒ(2))$. This means you replace $x$ in $ƒ(x) = 2x - 3$ with the value you found for $ƒ(2)$.

Write the expression for $ƒ(ƒ(2))$ as $2 \times ƒ(2) - 3$, where $ƒ(2)$ is the value you calculated in step 2.

Simplify the expression from step 4 to get the final value of $(ƒ \circ ƒ)(2)$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). In this problem, (ƒ∘ƒ)(2) means you first find ƒ(2), then apply ƒ again to that result.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function's formula and simplifying to find the output. For example, to find ƒ(2), replace x with 2 in ƒ(x) = 2x - 3.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. Both ƒ(x) = 2x - 3 and g(x) = -x + 3 are linear, which means their graphs are straight lines and their outputs change at a constant rate.

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