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The Algebra of Functions: Adding, Subtracting, Multiplying, Dividing, and Composition of Functions

Learn how to add, subtract, multiply, divide functions and compose functions with step-by-step examples. Simplify rational functions and understand conversion scales for practical interpretation.

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The Algebra of Functions: Adding, Subtracting, Multiplying, Dividing, and Composition of Functions

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  1. §0.3 The Algebra of Functions

  2. Section Outline • Adding Functions • Subtracting Functions • Multiplying Functions • Dividing Functions • Composition of Functions

  3. Adding Functions EXAMPLE Given and , express f (x) + g(x) as a rational function. SOLUTION f (x) + g(x) = Replace f (x) and g (x) with the given functions. Multiply to get common denominators. Evaluate. Add. Simplify the numerator.

  4. Adding Functions CONTINUED Evaluate the denominator. Simplify the denominator.

  5. Subtracting Functions EXAMPLE Given and , express f (x) - g(x) as a rational function. SOLUTION f (x) - g(x) = Replace f (x) and g (x) with the given functions. Multiply to get common denominators. Evaluate. Subtract. Simplify the numerator.

  6. Subtracting Functions CONTINUED Evaluate the denominator. Simplify the denominator.

  7. Multiplying Functions EXAMPLE Given and , express f (x)g(x) as a rational function. SOLUTION f (x)g(x) = Replace f (x) and g (x) with the given functions. Multiply the numerators and denominators. Evaluate.

  8. Dividing Functions EXAMPLE Given and , express [f (x)]/[g(x)] as a rational function. SOLUTION f (x)/g(x) = Replace f (x) and g (x) with the given functions. Rewrite as a product (multiply by reciprocal of denominator). Multiply the numerators and denominators. Evaluate.

  9. Composition of Functions EXAMPLE (Conversion Scales) Table 1 shows a conversion table for men’s hat sizes for three countries. The function converts from British sizes to French sizes, and the function converts from French sizes to U.S. sizes. Determine the function h(x) = f (g(x)) and give its interpretation. SOLUTION h(x) = f (g(x)) This is what we will determine. In the function f, replace each occurrence of x with g(x). Replace g(x) with 8x + 1.

  10. Composition of Functions CONTINUED Distribute. Multiply. Therefore, h(x) = f (g(x)) = x + 1/8. Now to determine what this function h(x) means, we must recognize that if we plug a number into the function, we may first evaluate that number plugged into the function g(x). Upon evaluating this, we move on and evaluate that result in the function f (x). This is illustrated as follows. g(x) f (x) British French French U.S. h(x) Therefore, the function h(x) converts a men’s British hat size to a men’s U.S. hat size.

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