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

Verified step by step guidance

1. Understand that the notation \((ƒ \circ g)(-2)\) means the composition of functions \(ƒ\) and \(g\) evaluated at \(-2\). This is written as \(ƒ(g(-2))\), which means you first find \(g(-2)\) and then plug that result into \(ƒ\).
2. Calculate \(g(-2)\) by substituting \(x = -2\) into the function \(g(x) = -x + 3\). So, \(g(-2) = -(-2) + 3\).
3. Simplify the expression for \(g(-2)\) to find its value. This will give you the input to use in the next step.
4. Next, substitute the value you found for \(g(-2)\) into the function \(ƒ(x) = 2x - 3\). So, calculate \(ƒ(g(-2)) = 2 \times g(-2) - 3\).
5. Simplify the expression for \(ƒ(g(-2))\) to find the final value of the composition \((ƒ \circ g)(-2)\).

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). To evaluate (f∘g)(-2), first find g(-2), then substitute that value into f. This process combines two functions into a single operation.

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(-2), replace x with -2 in g(x) = -x + 3 and simplify.

Linear Functions

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