• A function of the form 푦 = 푎푥푏– a is a real number– b is a rational number
• Let 푓(푥) = 3푥 ⁄ and 푔(푥) = 2푥 ⁄ . Find the following:– The sum
– The difference
– The domains
• Let 푓(푥) = 4푥 ⁄ and 푔(푥) = 푥 ⁄ . Find the following:– The product
– The quotient
– The domains
• The composition of the function 푓 with the function 푔 is:– ℎ(푥) = 푓(푔(푥))– The domain of h is the set of all x-values such that x is in
the domain of 푔 and 푔(푥)is in the domain of 푓– Need to pay attention to the order of functions when they
• Let 푓(푥) = 2푥 and 푔(푥) = 푥2 − 1. find the following:– 푓(푔(푥))
– The domains
• Let 푓(푥) = 푥 and 푔(푥) = 푥 + 1. Find the following:– 푓(푔(푥))
– The domains
• You do an experiment on bacteria and find that the growth rate G of the bacteria can be modeled by 퐺 푡 = 82푡 . , and that the death rate D is 퐷(푡) = 10.8푡 . , where t is the time in hours. Find an expression for the number N of bacteria living at time t.
• A computer catalog offers computers at a savings of 15% off the retail price. At the end of the month, it offers an additional 10% off its own price.– Use composition of function to find the total percent
– What would be the sale price of a $899 computer?
• P418: 12 – 51 x 3s, 54 – 56
• To find inverses of linear functions.• To find inverses of nonlinear functions.
• An Inverse Function maps the output values back to their original input values– So the range becomes the domain and the domain
becomes the range (or output to input and vice versa)– In other words, the x and y trade places– The graphs are always reflected over the line 푦 = 푥.
• Find the inverse of 푦 = −3푥 + 6.
• Function f and g are inverses of each other provided:– 푓(푔(푥)) = 푥– 푔(푓(푥)) = 푥
• The function g is denoted by 푓 , read as “f inverse”
• Verify that this function and your answer are inverses.
• A model for a salary is 푆 = 10.50ℎ + 50, where S is the total salary (in dollars) for one week and h is the number of hours worked.– Find the inverse function for the model.
– If a person’s salary is $533, how many hours does the person work?
• A model for a telephone bill is 푇 = 0.05푚 + 29.95,where T is the total bill, and m is the number of minutes used.– Find the inverse model.
– If the total bill is $54.15, how many minutes were used?
• Find the inverse of the function 푓(푥) = 푥
• Find the inverse of the function 푓(푥) = 푥 , 푥 ≥ 0
• If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.
• Consider the function 푓(푥) = 2푥 − 4. Determine whether the inverse of f is a function and then find the inverse.
• The volume of a sphere is given by 푉 = 휋푟3, where V is the volume and r is the radius. Write the inverse function that gives the radius as a function of the volume. Then determine the radius of a volleyball given that its volume is about 293 cubic inches.
• The volume of a cylinder with height 10 feet is given by 푉 = 10휋푟2, where V is the volume, and r is the radius. Write the inverse model that gives the radius as a function of the volume. Then determine the radius of a cylinder given that its volume is 1050 cubic feet.
• P426: 15 – 30 x 3s, 36 – 51 x 3s, 58, 62
• Graph 푦 = −2푥 + 3
• Daily Agenda:– Grade homework– Go over quiz…– 7.5 notes / assignment– 7.6 notes / assignment
Graphing Square root and Cube Root Functions
• To graph square root and cube root functions.• To use square root and cube root functions to find
• Look at P431 and find tables for x = 0, 1 for the square roots and tables for x = -1, 0, 1 for the cube roots.
• Activity: Investigating Graphs of Radical Functions– P431
• To graph 푦 = 푎 푥 − ℎ + 푘 or 푦 = 푎 푥 − ℎ + 푘, follow these steps.– Step 1: Sketch the graph of 푦 = 푎 푥 or 푦 = 푎 푥– Step 2: Shift the graph h units horizontally and k units
• Describe how to obtain the graph of 푦 = 푥 − 2 +1 from the graph of 푦 = 푥.
• Graph 푦 = 2 푥 + 4 − 1
• Describe how to obtain the graph of 푦 = 푥 − 3 + 2from the graph of 푦 = 푥. Then graph.
• Graph 푦 = −2 푥 − 3 + 2.
• Graph 푦 = −2 푥 − 1 + 1.
• State the domain and range of the function in all of the examples.
• A model for the period of a simple pendulum as
measured in time units is given by 푇 = 2휋 , where
푇 is the time in seconds, 퐿 is the length of the pendulum in feet, and 푔 is 32 ft/sec2. Use a graphing calculator to graph the model. Then use the graph to estimate the period of a pendulum that is 3 feet long.
• The length of a whale can be modeled by 퐿 =21.04 푊, where 퐿 is the length in feet, and 푊 is the weight in tons. Graph the model, then use the graph to find the weight of a what that is 60 ft long.