Exponential Functions and Equations

Introduction to Exponential Functions

100

What is the definition of an exponential function?

An exponential function is a mathematical function in which the variable appears in the exponent

200

Write the general notation for an exponential function.

The general notation for an exponential function is f(x) = a * b^x, where a is the initial value, b is the base, and x is the input variable.

300

Solve the exponential equation: 3^(2x) = 27

x=3/2

400

Solve the exponential equation: 2^(x + 1) = 8

x=2

500

Give an example of an application of exponential functions.

One example of an application of exponential functions is compound interest, where the amount of money in an investment grows exponentially over time.

Solving Inverse square roots

100

Solve for the inverse of y = x²

What is f^-1(x) = ±√(x)

200

Solve for the inverse of y = x² + 2

What is f^-1(x) = ±√(x - 2)

300

Solve for the inverse of y = -2x² - 4.

What is f^-1(x) = ±√( (x + 4) / -2)

400

Solve for the inverse of y = 5x² - 1

What is f^-1(x) = ±√((x + 1) / 5)

500

Solve for the inverse of y = 2(x² + 1)

What is f^-1(x) = ±√((x - 2) / 2)

solving multi-step inverses

100

Solve for the inverse of y = 2(x + 1)

What is f^-1(x) = (x - 2) / 2

200

Solve for the inverse of y = 3(2x + 1)

What is f^-1(x) = (x - 3) / 6

300

Solve for the inverse of y = 2(x + 1) - 3x

What is f^-1(x) = (x - 2) / -1

400

Solve for the inverse of y = 5x² + 4x² - 6

What is f^-1(x) = ±√( (x + 6) / 9)

500

Solve for the inverse of y = 5(x² + 2) - 2(x² + 1)

±√( (x - 8) / 3)

Exponential Growth and Decay Problems

100

The value of a smartphone depreciates at a rate of 15% per year. If the initial value is $1,000, find the smartphone's value after 3 years.

614.125

200

The value of a computer depreciates at a rate of 8% per year. If the initial value is $2,500, find the computer's value after 7 years.

1394.62

300

The population of rabbits doubles every 6 months. If there are initially 100 rabbits, how many will there be after 2 years? Applications of Exponential Growth and Decay

1600

400

The population of a city is currently 500,000, and it is growing exponentially at a rate of 3% per year. Estimate the population after 20 years.

903,055.62

500

Medication has a half-life of 6 hours. If a patient is given a 200 mg dose, how much will remain in their system after 24 hours?

12.5

Solving Exponential Equations

100

5^5y-1= 5³

y=4/5

200

(1/4)^x=8

-3/2

300

5^3x-2=125^2x

-2/3