Systems of linear equations

Precalc

(1) The amount of garbage, G, produced by a city with population p is given by $G = f(p)$. G is measured in tons per week, and p is measured in thousands of people. The town of Tola has a population of §§V0(1,100,1)§§,000 and produces §§V1(1,20,0.5)§§ tons of garbage each week. Express this information in terms of the function f: $f(§§V2(1,100,1)§§) = §§V3(1,20,0.5)§§$ Explain the meaning of the statement $f(§§V4(1,10,1)§§) = §§V5(1,5,0.1)§§$: This statement means that a city with a population of §§V4(1,10,1)§§,000 people produces §§V5(1,5,0.1)§§ tons of garbage per week. (2) Let $f(t)$ be the number of ducks in a lake $t$ years after 1990. Explain the meaning of each statement: $f(§§V6(1,15,1)§§) = §§V7(10,50,1)§§$ This statement means that in the year 1990 + §§V6(1,15,1)§§ = §§V8(1991,2005,1)§§, there were §§V7(10,50,1)§§ ducks in the lake. $f(§§V9(§§V6(1,15,1)§§ + 1, 20, 1)§§) = §§V10(§§V7(10,50,1)§§ + 5, 60, 1)§§$ This statement means that in the year 1990 + §§V9(§§V6(1,15,1)§§ + 1, 20, 1)§§ = §§V11(§§V8(1991,2005,1)§§ + 1, 2010, 1)§§, there were §§V10(§§V7(10,50,1)§§ + 5, 60, 1)§§ ducks in the lake. (3) Let $h(t)$ be the height above ground, in feet, of a rocket $t$ seconds after launching. Explain the meaning of each statement: $h(§§V12(1,5,1)§§) = §§V13(100,300,10)§§$ This statement means that §§V12(1,5,1)§§ seconds after the rocket was launched, its height above the ground was §§V13(100,300,10)§§ feet. $h(§§V14(§§V12(1,5,1)§§ + 1, 10, 1)§§) = §§V15(§§V13(100,300,10)§§ + 50, 400, 10)§§$ This statement means that §§V14(§§V12(1,5,1)§§ + 1, 10, 1)§§ seconds after the rocket was launched, its height above the ground was §§V15(§§V13(100,300,10)§§ + 50, 400, 10)§§ feet. (4) The number of cubic yards of dirt, D, needed to cover a garden with area a square feet is given by $D = g(a)$. A garden with area §§V16(1000,10000,500)§§ ft² requires §§V17(10,100,5)§§ $yd^3$ of dirt. Express this information in terms of the function g: $g(§§V16(1000,10000,500)§§) = §§V17(10,100,5)§§$ Explain the meaning of the statement $g(§§V18(10,200,10)§§) = §§V19(0.1,5,0.2)§§$: This statement means that a garden with an area of §§V18(10,200,10)§§ square feet requires §§V19(0.1,5,0.2)§§ cubic yards of dirt to be covered. (5) Show that the function $f(x) = 3(x - §§V20(-10,10,1)§§)^2 + §§V21(1,20,1)§§$ is not one-to-one. To show that the function $f(x) = 3(x - §§V20(-10,10,1)§§)^2 + §§V21(1,20,1)§§$ is not one-to-one, we need to find two different values of $x$ that produce the same value of $f(x)$. Consider $x_1 = §§V20(-10,10,1)§§ + §§V22(1,5,1)§§$ and $x_2 = §§V20(-10,10,1)§§ - §§V22(1,5,1)§§$. Since §§V22(1,5,1)§§ is a positive integer, $x_1 \neq x_2$. Now let's evaluate $f(x_1)$ and $f(x_2)$: $f(x_1) = 3((§§V20(-10,10,1)§§ + §§V22(1,5,1)§§) - §§V20(-10,10,1)§§)^2 + §§V21(1,20,1)§§ = 3(§§V22(1,5,1)§§)^2 + §§V21(1,20,1)§§ = 3 \cdot §§V23(1,25,1)§§ + §§V21(1,20,1)§§ = §§V24(3,75,3)§§ + §§V21(1,20,1)§§$ $f(x_2) = 3((§§V20(-10,10,1)§§ - §§V22(1,5,1)§§) - §§V20(-10,10,1)§§)^2 + §§V21(1,20,1)§§ = 3(-§§V22(1,5,1)§§)^2 + §§V21(1,20,1)§§ = 3(§§V22(1,5,1)§§)^2 + §§V21(1,20,1)§§ = 3 \cdot §§V23(1,25,1)§§ + §§V21(1,20,1)§§ = §§V24(3,75,3)§§ + §§V21(1,20,1)§§$ Since $f(x_1) = f(x_2)$ but $x_1 \neq x_2$, the function $f(x) = 3(x - §§V20(-10,10,1)§§)^2 + §§V21(1,20,1)§§$ is not one-to-one. This is because it's a quadratic function, and parabolas are symmetric around their vertex.