Performing Function Operations

5.5 Performing Function Operations Operations on Functions Let f and g be any two functions. A new function can be defined by performing any of the four basic operations on f and g. Operation Definition Example: f(x) = 5x, g(x) = x + 2 Addition (f + g)(x) = f(x) + g(x) (f + g)(x) = 5x + (x + 2) = 6x + 2 Subtraction (f - g)(x) = f(x) - g(x) (f - g)(x) = 5x - (x + 2) = 4x - 2 Multiplication (fg)(x) = f(x) * g(x) (fg)(x) = 5x * (x + 2) = 5x² + 10x Division (f/g)(x) = f(x) / g(x) (f/g)(x) = 5x / (x + 2)   The domains of the sum, difference, product, and quotient functions consist of the x-values that are in the domains of both f and g. Additionally, the domain of the quotient does not include x-values for which g(x) = 0.   B. EXAMPLES 1. Adding Two Functions a. Let f(x) = 3√x and g(x) = -10√x. Find (f + g)(x) and state the domain. Then evaluate the sum when x = 4. (f + g)(x) = f(x) + g(x) = 3√x - 10√x = -7√x Domain: x ≥ 0 (because you cannot take the square root of a negative number) Evaluate x = 4: -7√4 = -7(2) = -14 b. f(x) = 4x³ + 12, g(x) = 2x² - 3x³ + 9, x = 2 f(x) + g(x) = (4x³ + 12) + (2x² - 3x³ + 9) = x³ + 2x² + 21 Domain: All Real #s (no radicals, no even index or variable in denominator) Evaluate x = 2: (2)³ + 2(2)² + 12 = 8 + 2(4) + 12 = 28 2. Subtracting Two Functions a. Let f(x) = 3x³ - 2x² + 5 and g(x) = x³ - 3x² + 4x - 2. Find (f - g)(x) and state the domain. Then evaluate the difference when x = -2. (f - g)(x) = f(x) - g(x) = (3x³ - 2x² + 5) - (x³ - 3x² + 4x - 2) = 3x³ - 2x² + 5 - x³ + 3x² - 4x + 2 = 2x³ + x² - 4x + 7 Domain: All Real #s (because no domain restrictions) Evaluate x = -2: 2(-2)³ + (-2)² - 4(-2) + 7 = 2(-8) + 4 + 8 + 7 = -16 + 19 = 3 b. f(x) = -1/2√x, g(x) = 9/2√x; x = -1000 f(x) - g(x) = -1/2√x - 9/2√x = -10/2√x = -5√x Domain: All Real #s (odd index, okay for x-values to be positive or negative) Evaluate x = -1000: -5√-1000 = -5(-10) = 50 3. Multiplying Two Functions a. Let f(x) = x² and g(x) = √x. Find (fg)(x) and state the domain. Then evaluate the product when x = 9. (fg)(x) = f(x)g(x) = x² * √x = x² * x^(1/2) = x^(2+1/2) = x^(5/2) or (√x)⁵ Domain: x ≥ 0 (even index) Evaluate x = 9: (9)^(5/2) = (√9)⁵ = (3)⁵ = 243 b. f(x) = 36x^(7/4), g(x) = 4x^(1/2); x = 16 f(x)g(x) = 36x^(7/4) * 4x^(1/2) = 144x^(7/4+1/2) = 144x^(9/4) or 144(√x)⁹ Domain: x ≥ 0 (even index) Evaluate x = 16: 144 (√16)⁹ = 144 (2)⁹ = 144 (512) = 73,728 4. Dividing Two Functions a. Let f(x) = 6x⁴ and g(x) = x³/4. Find (f/g)(x) and state the domain. Then evaluate the quotient when x = 16. (f/g)(x) = f(x)/g(x) = 6x⁴ / x³/4 = 6x^(4 - 3/4) = 6x^(13/4) or 6√4 Domain: x ≥ 0 (even index) Evaluate x = 16: 6(16)^(13/4) = 6√4 = 6(2)¹³ = 8192 b. f(x) = 0.25x³, g(x) = -4x^(3/2); x = 1 (f/g)(x) = f(x)/g(x) = (0.25x³) / (-4x^(3/2)) = -0.0625x^(3 - 3/2) = -0.0625x^(3/2) Domain: x > 0 (even index and variable in denominator cannot divide by zero! x ≠ 0) Evaluate x = 1: -0.0625 (1)^(3/2) = -0.0625 = -0.0625 5. The variable x represents the number of pages of a textbook 1 to be printed. The cost C to print the textbook can be modeled by the equation C(x) = 0.2x² + 10. The selling price P of the textbook can be modeled by the equation P(x) = 0.05x² + 20.   a. Find (P - C)(x). P(x) - C(x) = (0.05x² + 20) - (0.2x² + 10) = 0.05x² + 20 - 0.2x² - 10 = -0.15x² + 10 b. Explain what (P - C)(x) represents. P(x) - C(x) = Selling Price - Cost to print textbook = The amount of profit from the sale of each textbook.