Math  /  Algebra

Question9. Consider the functions f(x)=3x,g(x)=x3f(x)=3 x, g(x)=x^{3}, and h(x)=3xh(x)=3^{x}. a) Graph each function. b) Make a list of the key features for each function, as in Investigate 2, step 2 b). Organize the information in a table. c) Identify key features that are common to each function. d) Identify key features that are different for each function. e) How do the instantaneous rates of change compare for these three functions?
10. An influenza virus is spreading through a school according to the function N=10(2)tN=10(2)^{t}, where NN is the number of people infected and tt is the time, in days. a) How many people have the virus at each time? i) initially, when t=0t=0 ii) after 1 day iii) after 2 days iv) after 3 days b) Graph the function. Does it appear to be exponential? Explain your answer. c) Determine the average rate of change between day 1 and day 2 . d) Estimate the instantaneous rate of change after i) 1 day ii) 2 days e) Explain why the answers to parts c) and d) are different. Use the functions f(x)=4xf(x)=4^{x} and g(x)=(12)xg(x)=\left(\frac{1}{2}\right)^{x} to answer questions 11 to 18.
11. a) Sketch a graph of ff. b) Graph the line y=xy=x on the same grid. c) Sketch the inverse of ff on the same grid by reflecting ff in the line y=xy=x.
12. Identify the key features of ff. a) domain and range b) xx-intercept, if it exists c) yy-intercept, if it exists

Studdy Solution

STEP 1

1. We are dealing with multiple functions and need to analyze their properties.
2. The functions are f(x)=3x f(x) = 3x , g(x)=x3 g(x) = x^3 , h(x)=3x h(x) = 3^x , and N=10(2)t N = 10(2)^t .
3. We need to graph functions, identify key features, and compare rates of change.
4. We need to evaluate the function N=10(2)t N = 10(2)^t at specific times.
5. We need to sketch and analyze the function f(x)=4x f(x) = 4^x and its inverse.

STEP 2

1. Graph each function and identify key features.
2. Compare key features of the functions.
3. Analyze the function N=10(2)t N = 10(2)^t at specific times.
4. Graph the function N=10(2)t N = 10(2)^t and analyze its properties.
5. Determine average and instantaneous rates of change for N=10(2)t N = 10(2)^t .
6. Sketch and analyze the function f(x)=4x f(x) = 4^x and its inverse.

STEP 3

Graph the functions f(x)=3x f(x) = 3x , g(x)=x3 g(x) = x^3 , and h(x)=3x h(x) = 3^x .

STEP 4

Identify key features for each function, such as domain, range, intercepts, and behavior.

STEP 5

Create a table to organize the key features of each function.

STEP 6

Identify common key features among the functions.

STEP 7

Identify different key features for each function.

STEP 8

Evaluate N=10(2)t N = 10(2)^t at t=0,1,2,3 t = 0, 1, 2, 3 .

STEP 9

Graph the function N=10(2)t N = 10(2)^t and determine if it is exponential.

STEP 10

Calculate the average rate of change between day 1 and day 2 for N=10(2)t N = 10(2)^t .

STEP 11

Estimate the instantaneous rate of change after 1 day and 2 days for N=10(2)t N = 10(2)^t .

STEP 12

Explain the difference between average and instantaneous rates of change.

STEP 13

Sketch the graph of f(x)=4x f(x) = 4^x and the line y=x y = x .

STEP 14

Sketch the inverse of f(x)=4x f(x) = 4^x by reflecting it in the line y=x y = x .

STEP 15

Identify key features of f(x)=4x f(x) = 4^x , including domain, range, and intercepts.

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