Applied Algebra: Modeling and functions

Subsection Key Concepts

1. Inverse Functions.

   If the inverse of a function \(f\) is also a function, then the inverse is denoted by the symbol \(f^{-1}\text{,}\) and

   \begin{equation*} f^{-1} (b) = a~~ \text{ if and only if } ~~f(a) = b \end{equation*}

2. We can make a table of values for the inverse function, \(f^{-1}\text{,}\) by interchanging the columns of a table for \(f\text{.}\)

3. If a function is defined by a formula in the form \(y = f (x)\text{,}\) we can find a formula for its inverse function by solving the equation for \(x\) to get \(x = f^{-1}(y)\text{.}\)

4. The inverse function \(f^{-1}\) undoes the effect of the function \(f\text{,}\) that is, if we apply the inverse function to the output of \(f\text{,}\) we return to the original input value.

5. If \(f^{-1}\) is the inverse function for \(f\text{,}\) then \(f\) is also the inverse function for \(f^{ -1}\text{.}\)

6. The graphs of \(f\) and its inverse function are symmetric about the line \(y = x\).

7. Horizontal line test: If no horizontal line intersects the graph of a function more than once, then the inverse is also a function.

8. A function that passes the horizontal line test is called one-to-one.

9. The inverse of a function \(f\) is also a function if and only if \(f\) is one-to-one.

10. We define the logarithmic function, \(g(x) = \log_{b} (x)\text{,}\) which takes the log base \(b\) of its input values. The log function \(g(x) = \log_{b} (x)\) is the inverse of the exponential function \(f (x) = b^x\text{.}\)

11. Because \(f (x) = b^x\) and \(g(x) = \log_{b} (x)\) are inverse functions for \(b\gt 0, ~b\ne 1\text{,}\)

    \begin{equation*} \log_{b} (b^x) = x~~\text{, for all }x~~\text{ and }~~~b^{\log_{b} (x)} = x~~\text{, for }x\gt 0 \end{equation*}

12. Logarithmic Functions \(y = \log_{b}(x)\).

    1. Domain: all positive real numbers
    2. Range: all real numbers
    3. \(x\)-intercept: \((1, 0)\)
    4. \(y\)-intercept: none
    5. Vertical asymptote at \(x = 0\)
    6. The graphs of \(y = \log_{b}(x)\) and \(y = b^x\) are symmetric about the line \(y = x\text{.}\)

13. A logarithmic equation is one where the variable appears inside of a logarithm. We can solve logarithmic equations by converting to exponential form.

14. Steps for Solving Logarithmic Equations.

    1. Use the properties of logarithms to combine all logs into one log.
    2. Isolate the log on one side of the equation.
    3. Convert the equation to exponential form.
    4. Solve for the variable.
    5. Check for extraneous solutions.

15. The natural base is an irrational number called \(e\text{,}\) where

    \begin{equation*} e\approx 2.71828182845 \end{equation*}

16. The natural exponential function is the function \(f (x) = e^x\text{.}\) The natural log function is the function \(g(x) = \ln (x) = \log_e (x)\text{.}\)

17. Conversion Formulas for Natural Logs.

    \begin{equation*} \blert{y = \ln (x)} ~~\text{ if and only if } ~~ \blert{e^y = x} \end{equation*}

18. Properties of Natural Logarithms.

    If \(x, y \gt 0\text{,}\) then

    1. \(\displaystyle \ln(xy) = \ln(x) + \ln(y)\)
    2. \(\displaystyle \ln\left(\dfrac{x}{y}\right) = \ln (x) - \ln (y)\)
    3. \(\displaystyle \ln(x^k) = k \ln (x) \)

    also

    \begin{equation*} \ln (e^x) = x\text{, for all }x\text{ and }e^{\ln(x)}=x\text{, for }x \gt 0 \end{equation*}

19. We use the natural logarithm to solve exponential equations with base \(e\text{.}\)

20. Exponential Growth and Decay.

    The function

    \begin{equation*} P(t) = P_0 e^{kt} \end{equation*}

    describes exponential growth if \(k \gt 0\text{,}\) and exponential decay if \(k \lt 0\text{.}\)

21. Continuous compounding: The amount accumulated in an account after \(t\) years at interest rate \(r\) compounded continuously is given by

    \begin{equation*} A(t) = Pe^{rt} \end{equation*}

    where \(P\) is the principal invested.

22. A log scale is useful for plotting values that vary greatly in magnitude. We plot the log of the variable, instead of the variable itself.

23. A log scale is a multiplicative scale: Each increment of equal length on the scale indicates that the value is multiplied by an equal amount.

24. The pH value of a substance is defined by the formula

    \begin{equation*} \text{pH}=-\log_{}\left([H^+]\right) \end{equation*}

    where \([H^+]\) denotes the concentration of hydrogen ions in the substance.

25. The loudness of a sound is measured in decibels, \(D\text{,}\) by

    \begin{equation*} D=10 \log_{}\left(\frac{I}{10^{-12}}\right) \end{equation*}

    where \(I\) is the intensity of its sound waves (in watts per square meter).

26. The Richter magnitude, \(M\text{,}\) of an earthquake is given by

    \begin{equation*} M=\log_{}\left(\frac{A}{A_0} \right) \end{equation*}

    where \(A\) is the amplitude of its seismographic trace and \(A_0\) is the amplitude of the smallest detectable earthquake.

27. A difference of \(K\) units on a logarithmic scale corresponds to a factor of \(10^K\) units in the value of the variable.

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