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Notes
Questions
Find the inverse of the following exponential functions.
\(\textbf{1)}\) Find the inverse of \( f(x)=10^x \)
\(\textbf{2)}\) Find the inverse of \( f(x)=2^{x} \)
\(\textbf{3)}\) Find the inverse of \( f(x)=2^{x-1}-2 \)
\(\textbf{4)}\) Find the inverse of \( f(x)=2^{x+3}+1 \)
\(\textbf{5)}\) Find the inverse of \( f(x)=e^{3x}+1 \)
\(\textbf{6)}\) Find the inverse of \( f(x)=-e^x+2 \)
\(\textbf{7)}\) Find the inverse of \( f(x)=4e^{-x} \)
\(\textbf{8)}\) Find the inverse of \( f(x)=-2e^{-x}+1 \)
\(\textbf{9)}\) Find the inverse of \( f(x)=10^{x}-3 \)
\(\textbf{10)}\) Find the inverse of \( f(x)=2e^{x+5} \)
See Related Pages\(\)
\(\bullet\text{ Exponential Functions}\)
\(\,\,\,\,\,\,\,\,y=a(b)^x\)
\(\bullet\text{ Compound Interest}\)
\(\,\,\,\,\,\,\,\,A=P\left(1+\frac{r}{n} \right)^{nt}\,\,\,\,\,\,\,\,A=P e^{rt}\)
\(\bullet\text{ Half Life Questions}\)
\(\,\,\,\,\,\,\,\,A_t=A_0e^{kt}\)
\(\bullet\text{ Graphing Exponential Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=2^{x-1}-2\,\, \)
\(\bullet\text{ Inverse of Exponential Functions}\)
\(\,\,\,\,\,\,\,\,f^{-1}(x)=\log_5(x-1)\)
In Summary
An inverse function in general is a function that “undoes” the process of another function. The way to undo an exponential function is called a logarithm. Logarithms are used to solve equations involving exponential functions. Exponential functions are used in the field of mathematics to handle real world problems involving exponential growth and decay, such as those found in population dynamics or radioactivity. Exponential Functions and Logarithms are typically covered in an algebra class in high school.
Real world examples of Exponential Functions
Radioactive decay: The rate at which a radioactive isotope decays is often modeled with an exponential function. The inverse of this function can be used to find the time it takes for a certain amount of the isotope to decay.
Population growth: The rate at which a population grows can also be modeled with an exponential function. The inverse of this function can be used to find the time it takes for the population to reach a certain size.
Compound interest: The amount of money in an investment account that is earning compound interest grows exponentially over time. The inverse of this function can be used to find the time it takes for the account to reach a certain balance.
Spread of a disease: The rate at which a disease spreads through a population can be modeled with an exponential function. The inverse of this function can be used to find the time it takes for the disease to reach a certain number of people.
Sound attenuation: The intensity of a sound wave decreases exponentially with distance. The inverse of this function can be used to find the distance at which the sound wave has a certain intensity.
Math topics related to Exponential Functions
Logarithms: Logarithms are the inverse of exponential functions. There is a lot of different types of fun exercises that can be solved with logarithms.
Exponential growth and decay: Exponential functions can be used to model exponential growth or decay, such as in population growth or the half-life of a radioactive substance.
Compound interest: Compound interest is the interest that is earned on the principal of an investment, as well as on any previously earned interest.
Continuously compounded interest: Continuously compounded interest, which is the interest that is earned on an investment that mimics growth in the natural world.
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