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Precalculus - Functions

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    Study Guide Precalculus – Functions 1. Combining and Composing Functions In mathematics, we can create new functions by working with functions we already know. Sometimes this is as simple as adding or subtracting two functions. Other times, we combine them in more powerful ways, such as composition , where one function is placed inside another. 1. Arithmetic Combinations of Functions The easiest way to make a new function is by using basic arithmetic operations : • Addition • Subtraction • Multiplication • Division These operations are applied directly to the formulas of the functions. Example 1 : Let (a) Find ((f - g)( - 1)) Step 1: Subtract the functions Step 2: Substitute (x = - 1)

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

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    Study Guide Function composition means plugging one function inside another. • 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 formula Step 3: Simplify

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    Study Guide Answer: (b) Find (g ∘ f)(4) This means find g(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 formula Step 3: Expand

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    Study Guide Step 4: Simplify Answer: 2. Inverse Functions You have actually been using inverse functions since your early algebra days — often without even realizing it! Anytime you “undo” an operation, such as subtracting after adding or dividing after multiplying, you are working with inverses. Now that your math skills have grown, let’s take a closer look at what inverse functions are , why they work , and how to find them . What Is an Inverse Function? An inverse function reverses the action of another function. If we start with a function ( f(x) ) , its inverse is written as: Key Idea • The domain of ( f(x) ) becomes the range of ( f - 1 (x) ) • The range of ( f(x) ) becomes the domain of ( f - 1 (x) ) In simple terms, inverse functions swap inputs and outputs . Ordered Pairs and Inverses If a point ((a, b)) belongs to ( f(x) ) , then the point ((b, a)) belongs to ( f - 1 (x) ) . Example

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