Math Combining Functions Key

Combining Functions Let f(x) = 2x + 3 and g(x) = 5x - 3. Find each value or expression. (f + g)(x) = f(x) + g(x) = 2x + 3 + 5x - 3 = 7x (f + g)(-3) = 7(-3) = -21 (f - g)(x) = f(x) - g(x) = 2x + 3 - (5x - 3) = 2x + 3 - 5x + 3 = -3x + 6 (f * g)(x) = f(x) * g(x) = (2x + 3)(5x - 3) = 10x^2 - 6x + 15 - 9 = 10x^2 + 9x - 9 (f * g)(-1) = 10(-1)^2 + 9(-1) - 9 = 10 - 9 - 9 = -8 (f ∘ g)(x) = f(g(x)) = f(5x - 3) = 2(5x - 3) + 3 = 10x - 6 + 3 = 10x - 3 (g ∘ f)(x) = g(f(x)) = g(2x + 3) = 5(2x + 3) - 3 = 10x + 15 - 3 = 10x + 12 (f ∘ f)(-5) = f(f(-5)) = f(2(-5) + 3) = f(-7) = 2(-7) + 3 = -11 9. The number of women and men age 16 and over employed each year in the U.S. can be modeled by the following equations. X is the number of years since 1994 and y is the number of people in thousands.   Women: w(x) = 1086.4x + 56,610 Men: m(x) = 999.2x + 66,450 a. Write a function that models the total number of men and women employed in the U.S. during this time. (w + m)(x) = 2085.6x + 123,060 b. Explain what (w - m)(x) represents. How many more women are employed than men, x years since 1994. 10. A ceramics store manufactures and sells coffee mugs. Revenue function: r(x) = 6.5x Cost function: c(x) = 0.75x + 1850 a) Profit equation: P(x) = r(x) - c(x) P(x) = 6.5x - (0.75x + 1850) P(x) = 5.75x - 1850 b) Profit of 5000 mugs: P(5000) = 5.75 * 5000 - 1850 = $26,900 For each pair of functions, find b(c(3)) and c(b(3)) b(x) = 2x - 1; c(x) = x² + 4 c(3) = 3² + 4 = 13 b(c(3)) = b(13) = 2(13) - 1 = 25 b(3) = 2(3) - 1 = 5 c(b(3)) = c(5) = 5² + 4 = 29 b(x) = x² + 4; c(x) = -x² - 1 c(3) = -3² - 1 = -10 b(c(3)) = b(-10) = (-10)² + 4 = 104 b(3) = 3² + 4 = 13 c(b(3)) = c(13) = -13² - 1 = -170 For each pair of functions, find f(g(x)) and g(f(x)) f(x) = 6x - 2; g(x) = x/3 f(g(x)) = f(x/3) = 6(x/3) - 2 = 2x - 2 g(f(x)) = g(6x - 2) = (6x - 2)/3 = 2x - 2/3 f(x) = x - 3; g(x) = x² f(g(x)) = f(x²) = x² - 3 g(f(x)) = g(x - 3) = (x - 3)² = x² - 6x + 9 For each pair of functions, find f(g(x)) and g(f(x)). f(x) = 5x; g(x) = (x+5)/10 f(g(x)) = f((x+5)/10) = 5((x+5)/10) = (5x+25)/10 = (x+5)/2 g(f(x)) = g(5x) = (5x+5)/10 = (x+1)/2 f(x) = 2(x^2) + x; g(x) = 5x f(g(x)) = f(5x) = 2(5x)^2 + 5x = 50x^2 + 5x g(f(x)) = g(2x^2 + x) = 5(2x^2 + x) = 10x^2 + 5x f(x) = x + 2; g(x) = x^2 + 3x + 4 f(g(x)) = f(x^2 + 3x + 4) = x^2 + 3x + 4 + 2 = x^2 + 3x + 6 g(f(x)) = g(x + 2) = (x + 2)^2 + 3(x + 2) + 4 = x^2 + 7x + 14 f(x) = 2x + 7; g(x) = 5x - 12 f(g(x)) = f(5x - 12) = 2(5x - 12) + 7 = 10x - 17 g(f(x)) = g(2x + 7) = 5(2x + 7) - 12 = 10x + 23 The surface area, S, of a sphere is a function of the radius, r, such that S(r) = 4πr^2. The radius is a function of the diameter, d, r(d) = 0.5d. Write and simplify an expression S(r(d)) the surface area as a function of the diameter. S(r(d)) = S(0.5d) = 4π(0.5d)^2 = πd^2 The weekly cost of producing x units in a manufacturing process is given by the function C(x) = 60x + 750. The number of units produced in t hours is given by x(t) = 50t. What does C(x(t)) represent? C(x(t)) = C(50t) = 60(50t) + 750 = 3000t + 750 C(x(t)) represents the cost of producing x units in t hours. Is the relation {(3, 1), (2, 1), (1, 3)} a function? Is its inverse relation a function? Original: Yes, it's a function. Inverse: {(1, 3), (1, 2), (3, 1)}: No, it's not a function. Is the relation {(2, 4), (6, 8), (10, 12)} a function? Is its inverse relation a function? Original: Yes, it's a function. Inverse: {(4, 2), (8, 6), (12, 10)}: Yes, it's a function. Graph each relation. Then, graph its inverse. {(2, 3), (3, 4), (-4, 5)} Original: Points plotted on a graph Inverse: {(3, 2), (4, 3), (5, -4)} Points plotted on the same graph y = 2x - 5 Original: Line graphed Inverse: (0, -5), (1, -3), (4, 3) Points plotted on the graph Show algebraically whether the given functions are inverses of each other. f(x) = 2x + 4; g(x) = (x - 4)/2 f(g(x)) = f((x - 4)/2) = 2((x - 4)/2) + 4 = x - 4 + 4 = x g(f(x)) = g(2x + 4) = (2x + 4 - 4)/2 = 2x/2 = x Yes, they are inverses. f(x) = 4x + 5; g(x) = x - 5 f(g(x)) = f(x - 5) = 4(x - 5) + 5 = 4x - 20 + 5 = 4x - 15 No, they are not inverses. Find the inverse of each function. f(x) = 0.25x y = 0.25x x = 0.25y 4x = y = f^(-1)(x) f(x) = 1/2x + 1 y = 1/2x + 1 x = 1/2y + 1 2(x - 1) = y^2 √(2x - 2) = y = f^(-1)(x) f(x) = 8x^3 y = 8x^3 x = 8y^3 (x/8)^(1/3) = y = f^(-1)(x) f(x) = x/3 - 4 y = x/3 - 4 x = y/3 - 4 3(x + 4) = y = f^(-1)(x)