Composition of linear functions

Learning Objectives Define function composition and correctly use the notation (f ∘ g)(x) and f(g(x)). Evaluate the composition of two linear functions at a specific numerical value. Determine the algebraic expression for the composition of two linear functions. Analyze whether the composition of linear functions is commutative by comparing (f ∘ g)(x) and (g ∘ f)(x). Decompose a given linear function into two simpler linear functions. Model and solve multi-step real-world problems using the composition of linear functions. Ever wonder how a website calculates the final price of an item after applying a percentage discount and then a fixed shipping fee? 🛒 You're actually using a composition of functions! This tutorial explores the composition of linear functions, a pro...

TermDefinitionExample Linear FunctionA function that can be written in the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept. Its graph is a straight line.f(x) = 2x - 3 is a linear function with a slope of 2 and a y-intercept of -3. Function CompositionThe process of applying one function to the result of another. The output of the first (inner) function becomes the input for the second (outer) function.If a function g(x) doubles a number and f(x) adds 1 to a number, the composition (f ∘ g)(3) would first double 3 to get 6, and then add 1 to get 7. Composition NotationThe composition of function 'f' with function 'g' is written as (f ∘ g)(x) or f(g(x)). It is read as 'f of g of x'.For f(x) = 5x and g(x) = x + 2, the com...

Composition of f with g (f \circ g)(x) = f(g(x)) To find the composition, substitute the entire expression for the inner function, g(x), into every instance of the variable 'x' in the outer function, f(x). Composition of g with f (g \circ f)(x) = g(f(x)) To find this composition, substitute the entire expression for the inner function, f(x), into every instance of the variable 'x' in the outer function, g(x). Composition of Two Linear Functions If f(x) = m_1x + b_1 and g(x) = m_2x + b_2, then (f \circ g)(x) is also a linear function with slope m_1m_2. The composition of two linear functions always results in another linear function. The new slope is the product of the original slopes, and the new y-intercept is m_1b_2 + b_1.