Precalculus - Functions | Mathematics

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Study GuidePrecalculus–Functions1.Combining and Composing FunctionsIn mathematics, we can createnew functionsby working with functions we already know.Sometimes this is as simple as adding or subtracting two functions. Other times, we combine them inmore powerful ways, such ascomposition, where one function is placed inside another.1. Arithmetic Combinations of FunctionsThe easiest way to make a new function is by usingbasic arithmetic operations:•Addition•Subtraction•Multiplication•DivisionThese operations are applied directly to the formulas of the functions.Example 1:Let(a) Find ((f-g)(-1))Step 1: Subtract the functionsStep 2: Substitute (x =-1)

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Study GuideAnswer:((f-g)(-1) = 0)(b)Find (f/g)(x)Step 1: Write the divisionStep 2: Factor the numeratorStep 3: SimplifyThis simplification is validonly if(x≠-1), because division by zero is not allowed.Answer:Important Note:The domain of this new function isall real numbers except (x =-1). This restriction comes from thedenominator, even though both original functions are defined for all real numbers.2. Composition of FunctionsNow let’s look at a different way to combine functions.What Is Function Composition?

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Study GuideFunction compositionmeans plugging one functioninsideanother.•Written as:•This means:1.Start with (x)2.Apply (g)3.Take that result and apply (f)Composition can also be written as:Example 2:Let(a)Find (f(g(x)))Step 1: Replace (x) in (f(x)) with (g(x))Step 2: Substitute into the formulaStep 3: Simplify

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Study GuideAnswer:(b)Find (g∘f)(4)This meansfindg(f(4)).Step 1: Find (f(4))Step 2: Substitute into (g(x))Answer:(c)Find ((f∘f)(x))Step 1: Replace (x) in (f(x)) with (f(x))Step 2: Substitute into the formulaStep 3: Expand

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Study GuideStep 4: SimplifyAnswer:2.Inverse FunctionsYou have actually been usinginverse functionssince your early algebra days—often without evenrealizing it! Anytime you “undo” an operation, such as subtracting after adding or dividing aftermultiplying, you are working with inverses.Now that your math skills have grown, let’s take a closer look atwhat inverse functions are,whythey work, andhow to find them.What Is an Inverse Function?Aninverse functionreverses the action of another function.If we start with a function(f(x)), its inverse is written as:Key Idea•Thedomainof(f(x))becomes therangeof(f-1(x))•Therangeof(f(x))becomes thedomainof(f-1(x))In simple terms, inverse functionsswap inputs and outputs.Ordered Pairs and InversesIf a point ((a, b)) belongs to(f(x)), then the point ((b, a)) belongs to(f-1(x)).Example

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