Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5.
$(\text{ƒ}\circ g)\left(4\right)$

Verified step by step guidance

1. Understand that the notation \((ƒ \circ g)(4)\) means you first apply the function \(g\) to 4, then apply the function \(ƒ\) to the result of \(g(4)\).
2. Calculate \(g(4)\) by substituting \(x = 4\) into the function \(g(x) = -x + 3\). This gives \(g(4) = -4 + 3\).
3. Simplify the expression for \(g(4)\) to find its value.
4. Next, take the result from \(g(4)\) and substitute it into the function \(ƒ(x) = 2x - 3\). This means you calculate \(ƒ(g(4)) = 2 \times g(4) - 3\).
5. Simplify the expression for \(ƒ(g(4))\) to find the final value of \((ƒ \circ g)(4)\).

Key Concepts

Function Composition

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

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 g(4), replace x with 4 in g(x) = -x + 3 and simplify. Accurate evaluation is crucial for solving composite functions.

Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are easy to evaluate and compose. Understanding their structure helps in quickly calculating values and interpreting function behavior.