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

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation \((f \circ g)(x)\) means \(f(g(x))\), which involves substituting \(g(x)\) into \(f(x)\).
2. Step 2: Start by finding \(g(-2)\). Substitute \(-2\) into the function \(g(x) = -x + 3\).
3. Step 3: Calculate \(g(-2) = -(-2) + 3\). Simplify this expression to find the value of \(g(-2)\).
4. Step 4: Use the result from Step 3 as the input for the function \(f(x)\). Substitute \(g(-2)\) into \(f(x) = 2x - 3\).
5. Step 5: Calculate \(f(g(-2)) = 2(g(-2)) - 3\). Simplify this expression to find the value of \((f \circ g)(-2)\).

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (ƒ∘g)(x) means to apply g first and then apply f to the result. This process requires substituting the output of g into the function f, which is essential for solving problems that involve multiple functions.

Evaluating Functions

Evaluating a function means finding the output value for a given input. For example, to evaluate g(-2), you substitute -2 into the function g(x) and calculate the result. This step is crucial in function composition, as the output of the first function becomes the input for the second function.

Linear Functions

Linear functions are mathematical expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Both ƒ(x) and g(x) in this problem are linear functions, which means their graphs are straight lines. Understanding their properties helps in evaluating and composing these functions effectively.