The regular price of a computer is x dollars. Let f(x)=x-400...

The regular price of a computer is $x$ dollars. Let $f(x)=x-400$ and $g(x)=0.75 x$ a. Describe what the functions $f$ and $g$ model in terms of the price of the computer. b. Find $(f \circ g)(x)$ and describe what this models in terms of the price of the computer. c. Repeatpart(b) for $(g \circ f)(x)$. d. Which composite function models the greater discount on the computer, $f \circ g$ or $g \circ f ?$ Explain. e. Find $f^{-1}$ and describe what this models in terms of the price of the computer.

The regular price of a computer is $x$ dollars. Let $f(x)=x-400$ and $g(x)=0.75 x$
a. Describe what the functions $f$ and $g$ model in terms of the price of the computer.
b. Find $(f \circ g)(x)$ and describe what this models in terms of the price of the computer.
c. Repeatpart(b) for $(g \circ f)(x)$.
d. Which composite function models the greater discount on the computer, $f \circ g$ or $g \circ f ?$ Explain.
e. Find $f^{-1}$ and describe what this models in terms of the price of the computer.

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Key Concepts

Inverse Functions

An inverse function reverses the operation of the original function. In the context of price modeling, finding the inverse function can be used to determine the original price from a given discounted price, thereby 'undoing' the applied discount.

Order of Operations in Compositions

The order in which functions are composed can lead to different results, especially when each function represents a distinct transformation. In price discount scenarios, applying a flat discount before a percentage discount may yield a different final price than applying the percentage discount first, highlighting the importance of sequence in composite functions.

Function Composition

Function composition is the process of applying one function to the result of another, symbolically written as (f ? g)(x) or (g ? f)(x). This concept illustrates how combining successive operations affects the final output, which in this context shows the cumulative effect of applying multiple discounts.

Function Modeling

This concept involves using functions to represent real-world relationships, such as price adjustments in a computer sale. Functions like f(x) and g(x) can model different types of discounts or price modifications applied to an item's original price.

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