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

Verified step by step guidance

1. 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 \(x=2\).
2. First, find the inner function value \(ƒ(2)\) by substituting \(x=2\) into the function \(ƒ(x) = 2x - 3\). This gives \(ƒ(2) = 2(2) - 3\).
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 \(ƒ(2)\).
4. Write the expression for \(ƒ(ƒ(2))\) as \(2 \times ƒ(2) - 3\), where \(ƒ(2)\) is the value you calculated in step 2.
5. Simplify the expression from step 4 to get the final value of \((ƒ \circ ƒ)(2)\).

Key Concepts

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.