Function Operations Worksheets Grade 12

1. Aroha is sketching the graph of the rational function formed by combining two polynomial functions: f(x) = x^3 - 9x and g(x) = x^2 - 3x. The graph of h(x) = (f/g)(x) has vertical asymptotes and a hole at certain x-values. Determine the coordinates (x, y) of the hole in the graph of h(x).
2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is C(t) = 5t²e^(-0.2t) mg/L, where t is time in hours. To determine the maximum concentration and when it occurs, the research team needs to find the critical points of this function. Find the derivative C'(t) and determine at what time the drug concentration reaches its maximum value.
3. Aroha is a marine ecologist studying two species of algae in a coastal ecosystem. The biomass of Species X after t days is modeled by f(t) = 12t^2 + 9t + 15 kilograms, and the biomass of Species Y is modeled by g(t) = 9t + 10 kilograms. Aroha wants to understand how the total biomass of both species combined changes over time, and also the ratio of the biomass of Species X to Species Y. Determine the combined biomass function h(t) = (f + g)(t) and the ratio function r(t) = (f / g)(t). Then find the total combined biomass when t = 10 days.

…and 4 more problems

1. Kaia is a wildlife ecologist studying two species of native birds in a forest reserve. The population of the kākā (in hundreds) after t years is modeled by f(t) = 3t² + 7t + 1, and the population of the kea (in hundreds) is modeled by g(t) = 5t + 9. To understand the combined population dynamics, Kaia defines the total population function T(t) = (f + g)(t). She also needs to find the ratio function R(t) = (f ÷ g)(t) to analyze how the kākā population compares to the kea population over time. Determine T(t) in simplified form, and find R(t) in simplified form, stating any restrictions on the domain for R(t) that make sense in this ecological context.
2. Liam is analyzing the graph formed by combining two functions on a coordinate plane: f(x) = x^2 - 25 and g(x) = 2x + 10. The graph of h(x) = (f/g)(x) has a vertical asymptote and a hole. Determine the coordinates (x, y) of the hole in the graph of h(x).
3. Emma is a civil engineer designing a suspension bridge. The main cable's height above the bridge deck is modeled by the quadratic function h(x) = 0.0005x² - 0.5x + 150, where x is the horizontal distance in meters from the west anchor. The vertical support cables are attached at regular intervals, and the tension in a support cable at position x is given by the linear function T(x) = 0.15x + 20 (in kilonewtons). The total vertical load on the bridge deck at any point x is the product of the cable height and the tension in the support cable, given by L(x) = h(x) × T(x). Determine the function L(x) that models the total vertical load on the bridge deck.

…and 3 more problems

1. Emma is a water quality analyst studying two pollutants in a river. The concentration of pollutant A (in parts per million) over time t (in years) is modeled by the function A(t) = 5t^2 + 20t, and the concentration of pollutant B is modeled by B(t) = 15t + 10. Emma needs to determine the function that represents the total concentration of both pollutants combined, T(t) = A(t) + B(t). She then plans to use the quotient function R(t) = T(t) / B(t) to analyze the ratio of total pollutants to pollutant B. Find both T(t) and R(t), and simplify R(t) as much as possible.
2. f(x) = 7x³ - 12x² + 17, g(x) = 2x³ + 7x - 22. Find (f + g)(x) = ?
3. f(x) = 2x² - 7x + 3, g(x) = x² + 2x - 5. Find (f + g)(x) = ?

…and 4 more problems