Introduction to Exponential Functions
What is the definition of an exponential function?
An exponential function is a mathematical function in which the variable appears in the exponent
Name one technique for solving exponential equations?
1) taking logarithms
2) using properties of exponents, and simplifying the equation to isolate the variable.
3) using the 1-to-1 property to eliminate bases
Define exponential growth and provide an example.
Exponential growth is a type of growth in which a quantity increases at an accelerating rate proportional to its current value. Example: Population growth in an idealized environment where resources are unlimited.
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
Find the Domain and Range of f(x)=5^(x)
Domain: (−∞,∞)
Range: (0,∞)
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.
How can logarithmic functions and equations be used to solve exponential equations?
Logarithmic functions and equations can be used to solve exponential equations by converting the equation into a logarithmic form and then using the properties of logarithms to simplify and solve for the variable.
Define exponential decay and provide an example.
Exponential decay is a type of decay in which a quantity decreases at a decelerating rate proportional to its current value. Example: Radioactive decay of a substance over time.
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
Find the Domain and Range of f(x)=6^(x) -5
Domain: (−∞,∞)
Range: (−5,∞)
Describe the general characteristics of the graph of an exponential.
Exponential graphs are curves that either increase or decrease with a horizontal asymptote.
Solve the exponential equation: 3^(2x) = 27.
x=3/2
What is half-life in the context of exponential decay?
Half-life is the time it takes for half of a substance to decay or for a quantity to decrease by half in the context of exponential decay.
The population of rabbits doubles every 6 months. If there are initially 100 rabbits, how many will there be after 2 years?
1600
Find the Domain and Range of f(x)=-2^(x) +1
Domain: (−∞,∞)
Range: (−∞,1)
Name two characteristics of exponential functions.
1) The domain is all real numbers. 2) The range is limited. 3) There is a horizontal asymptote. etc.
Solve the exponential equation: 2^(x + 1) = 8.
x=2
If an exponential function involves "doubling", how would that look in the equation of the function?
The base in the function would be 2.
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
Find the Domain and Range of f(x)=-3^(x) -5
Domain: (−∞,∞)
Range: (−∞,-5)
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.
Give an example of an application problem that involves solving exponential equations.
Determining the time it takes for a radioactive substance to decay to a certain level given its half-life.
Give an example of an application problem involving exponential growth or decay.
An example of an application problem involving exponential growth or decay is modeling the spread of a contagious disease in a population over time, considering factors such as transmission rate and recovery rate.
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
Find the Domain and Range f(x)=-3^(x)+5
Domain: (−∞,∞)
Range: (−∞,5)