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UNIT 4 LESSON 5 NOTES 2 Solving f(x) = g(x) Algebraically EXAMPLE: Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount? If x is the number of years and y is the weight, write an equation that represents the bunny’s weight (y) after a given number of years (x). Bunny: y = 3x + 5 Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount? If x is the number of years and y is the weight, write an equation that represents the cat’s weight (y) after a given number of years (x). Cat: y = 1x + 19 When two equations are equal to the same thing (or variable) then you can put them equal to each other and solve. Bunny: y = 3x + 5 3x + 5 = x + 19 -1x -x 2x + 5 = 19 -5 -5 2x = 14 x = 7 Cat: y = x + 19 The Bunny and Cat’s weight will be equal after 7 years. SOLVING F(X) = G(X) ALGEBRAICALLY Replace any function notation f(x), g(x), with y = Place the two functions equal to each other. Looking at what type of function you have, solve the equation using appropriate method. SOLVING DIFFERENT TYPES OF EQUATIONS Linear: 1. Get x to one side of equal sign. 2. If a constant is with x, add/subtract that value to both sides. 3. If a number is being multiplied by x, divide both sides by that number. Quadratic: 1. If there is an x2 and NO x term, get x2 by itself and take square root of both sides. 2. If there is an x2 AND an x term, get one side equal to zero, factor , put equal to zero and then solve for x. Absolute Value: 1. Get |x| by itself on one side of equal sign. 2. Make two equations and set equal to positive answer AND negative answer. 3. Solve for x. EXAMPLE 1: FIND THE SOLUTION TO F(X) = G(X). f(x) = 7x – 3 and g(x) = x + 15 (same as y = 7x – 3 and y = x + 15) Both are linear functions 7x – 3 = x + 17 6x – 3 = 15 6x = 18 x=3 EXAMPLE 2: FIND THE SOLUTION TO F(X) = G(X). f(x) = x2 – 4 Quadratic but NO ‘x’ term and g(x) = 45 x2 – 4 = 45 x2 = 49 x = 7 and -7 EXAMPLE 3: FIND THE SOLUTION TO F(X) = G(X). f(x) = x2 – 5x Quadratic - with x2 AND ‘x’ term and g(x) = 2x – 10 x2 – 5x = 2x – 10 x2 – 7x + 10 = 0 (x – 5)(x – 2) = 0 x = 5 and x = 2 EXAMPLE 4: FIND THE SOLUTION TO F(X) = G(X). f(x) = |x – 19| and g(x) = 22 |x – 19| = 22 Absolute Value Function x – 19 = 22 x – 19 = -22 x = 41 and x = -3