Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).
h(x) = |2x-5|

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with expressing the given function h(x) = |2x - 5| as a composition of two functions f(x) and g(x), such that h(x) = (f ∘ g)(x). This means h(x) = f(g(x)).
2. Step 2: Identify the inner function g(x). The expression inside the absolute value, 2x - 5, can be treated as the inner function. Let g(x) = 2x - 5.
3. Step 3: Identify the outer function f(x). The absolute value operation is applied to the result of g(x). Therefore, the outer function is f(x) = |x|.
4. Step 4: Verify the composition. Substitute g(x) into f(x) to ensure the composition matches h(x). f(g(x)) = f(2x - 5) = |2x - 5|, which is the original function h(x).
5. Step 5: Conclude that the functions are f(x) = |x| and g(x) = 2x - 5, and their composition satisfies h(x) = (f ∘ g)(x).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. If we have two functions f(x) and g(x), the composition is denoted as (f o g)(x) = f(g(x)). Understanding this concept is crucial for expressing the function h(x) as a composition of two simpler functions.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is essential in the given problem because h(x) = |2x - 5| requires us to consider how to express the linear transformation 2x - 5 in a way that can be composed with another function to yield the absolute value.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the context of the problem, recognizing that 2x - 5 is a linear function helps in identifying suitable functions f and g that can be composed to achieve the desired absolute value output.