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Over 1 month ago
Prompt Details
Model
Chat - GPT-3.5 (gpt-3.5-turbo)
Token size
138 ($0.00030 / call)
Example input
[science subject]: mathematics, [level of study/class]: 11th students , [insert country of study]: US, [chapter name]: exponential functions, [insert concepts]: determining the properties of exponential functions, solving exponential equations, and manipulating logarithms
Example output
As a mathematics teacher for 11th grade students in the US, I understand that your student is having difficulties with the chapter on exponential functions. To help improve their understanding, I recommend the following set of exercises:
Determine the properties of the function f(x) = 2^x:
a. Find the domain and range of the function.
b. Determine whether the function is increasing or decreasing.
c. Find the x- and y-intercepts of the function.
Solve the following exponential equation for x:
a. 3^(2x+1) = 27
b. 4^x = 1/16
Simplify the following logarithmic expressions:
a. log4(16) - log4(4)
b. 2log7(5) - log7(2)
For the correction, let's start with the first exercise. To find the domain and range of f(x) = 2^x, we know that the base is 2, which is positive and not equal to 1. This means that the domain of the function is all real numbers and the range is all positive numbers. To determine whether the function is increasing or decreasing, we take the derivative of the function, which is f'(x) = ln(2) * 2^x. Since ln(2) is positive, this means that the function is increasing. Finally, to find the x- and y-intercepts of the function, we set x and y equal to zero respectively. Therefore, the x-intercept is (0,1) and there is no y-intercept.
For the second exercise, we can solve 3^(2x+1) = 27 by first simplifying the right-hand side to 3^3. Then, we can equate the exponents on both sides to get 2x + 1 = 3. Solving for x gives us x = 1/2. Similarly, we can simplify 4^x = 1/16 to get 2^2x = 2^-4. Equating the exponents again, we get 2x = -4. Solving for x gives us x = -2.
Finally, for the third exercise, we can simplify log4(16) - log4(4) to get log4(4) = 1. For the second part, we can use the property loga(b^c) = cloga(b) to simplify 2log7(5) - log7(2) to log7(5^2) - log7(2) = log7(25/2).
I hope these exercises and corrections help improve your student's understanding of exponential functions. Don't hesitate to ask if you have any further questions or concerns.