Patterns and structure. The student applies mathematical processes to understand the connections among representations of functions and combinations of functions, including the constant function, 𝑓(𝑥) = 𝑥, 𝑓(𝑥) = 𝑥², 𝑓(𝑥) = √𝑥, 𝑓(𝑥) = 1/𝑥, 𝑓(𝑥) = 𝑥³, 𝑓(𝑥) = ³√𝑥, 𝑓(𝑥) = 𝑏 to the 𝑥 power, 𝑓(𝑥) = |𝑥|, and 𝑓(𝑥)=𝑙𝑜𝑔 𝑏𝑎𝑠𝑒-𝑏 (𝑥) where 𝑏 is 2, 10 or 𝑒; functions and their inverses; and key attributes of these functions.
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More specific sub-standards
compare and contrast the key attributes of a function and its inverse when it exists, including domain, range, maxima, minima, and intercepts, tabularly, graphically, and symbolically;
compare and contrast a function and possible functions that can be used to build it tabularly, graphically, and symbolically such as a quadratic function that results from multiplying two linear functions.
model a situation using function notation when the output of one function is the input of a second function such as determining a function ℎ(𝑥) = 𝑔(𝑓(𝑥)) = 1.06(0.8𝑥) for the final purchase price, ℎ(𝑥) of an item with price 𝑥 dollars representing a 20% discount, 𝑓(𝑥) = 0.8𝑥 followed by a 6% sales tax, 𝑔(𝑥) = 1.06𝑥; and
verify that two functions are inverses of each other tabularly and graphically such as situations involving compound interest and interest rate, velocity and braking distance, and Fahrenheit-Celsius conversions;
represent a resulting function tabularly, graphically, and symbolically when functions are combined or separated using arithmetic operations such as combining a 20% discount and a 6% sales tax on a sale to determine ℎ(𝑥), the total sale, 𝑓(𝑥) = 0.8𝑥, 𝑔(𝑥) = 0.06(0.8𝑥), and ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥);