10.1 Finding Composite and Inverse Functions

Composition of Functions: The composition of functions $f$ and $g,$ is written $f \circ g$ and is defined by

$(f \circ g) (x) = f (g (x))$

We read $f (g (x))$ as $f$ of $g$ of $x .$
Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
Inverse of a Function Defined by Ordered Pairs: If $f (x)$ is a one-to-one function whose ordered pairs are of the form $(x, y),$ then its inverse function $f^{−1} (x)$ is the set of ordered pairs $(y, x) .$
Inverse Functions: For every $x$ in the domain of one-to-one function $f$ and $f^{−1},$

$\begin{matrix} f^{−1} (f (x)) & = & x \\ f (f^{−1} (x)) & = & x \end{matrix}$
How to Find the Inverse of a One-to-One Function:
1. Step 1. Substitute y for $f (x) .$
2. Step 2. Interchange the variables x and y.
3. Step 3. Solve for y.
4. Step 4. Substitute $f^{−1} (x)$ for $y .$
5. Step 5. Verify that the functions are inverses.

10.2 Evaluate and Graph Exponential Functions

Properties of the Graph of

f (x) = a^{x} :

One-to-One Property of Exponential Functions:
For $a > 0$ and $a \neq 1,$

$If a^{x} = a^{y}, then x = y .$
How to Solve an Exponential Equation
1. Step 1. Write both sides of the equation with the same base, if possible.
2. Step 2. Write a new equation by setting the exponents equal.
3. Step 3. Solve the equation.
4. Step 4. Check the solution.
Compound Interest: For a principal, $P,$ invested at an interest rate, $r,$ for $t$ years, the new balance, $A,$ is
$\begin{matrix} A = P {(1 + \frac{r}{n})}^{n t} & when compounded n times a year. \\ A = P e^{r t} & when compounded continuously. \end{matrix}$
Exponential Growth and Decay: For an original amount, $A_{0}$ that grows or decays at a rate, $r,$ for a certain time $t,$ the final amount, $A,$ is $A = A_{0} e^{r t} .$

Properties of the Graph of

y = {log}_{a} x :

Decibel Level of Sound: The loudness level, $D$ , measured in decibels, of a sound of intensity, $I$ , measured in watts per square inch is $D = 10 log (\frac{I}{10^{−12}}) .$
Earthquake Intensity: The magnitude $R$ of an earthquake is measured by $R = log I,$ where $I$ is the intensity of its shock wave.

Properties of Logarithms

${log}_{a} 1 = 0 {log}_{a} a = 1$
Inverse Properties of Logarithms
- For $a > 0,$ $x > 0$ and $a \neq 1$
  
  $a^{{log}_{a} x} = x {log}_{a} a^{x} = x$
Product Property of Logarithms
- If $M > 0, N > 0, a > 0$ and $a \neq 1,$ then,
  
  ${log}_{a} M \cdot N = {log}_{a} M + {log}_{a} N$
  
  The logarithm of a product is the sum of the logarithms.
Quotient Property of Logarithms
- If $M > 0, N > 0, a > 0$ and $a \neq 1,$ then,
  
  ${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$
  
  The logarithm of a quotient is the difference of the logarithms.
Power Property of Logarithms
- If $M > 0, a > 0, a \neq 1$ and $p$ is any real number then,
  
  ${log}_{a} M^{p} = p {log}_{a} M$
  
  The log of a number raised to a power is the product of the power times the log of the number.

Properties of Logarithms Summary
If

M > 0, a > 0, a \neq 1

and

p

is any real number then,

Property	Base $a$	Base $e$
	${log}_{a} 1 = 0$	$ln 1 = 0$
	${log}_{a} a = 1$	$ln e = 1$
Inverse Properties	$\begin{matrix} a^{{log}_{a} x} = x \\ {log}_{a} a^{x} = x \end{matrix}$	$\begin{matrix} e^{ln x} = x \\ ln e^{x} = x \end{matrix}$
Product Property of Logarithms	${log}_{a} (M \cdot N) = {log}_{a} M + {log}_{a} N$	$ln (M \cdot N) = ln M + ln N$
Quotient Property of Logarithms	${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$	$ln \frac{M}{N} = ln M - ln N$
Power Property of Logarithms	${log}_{a} M^{p} = p {log}_{a} M$	$ln M^{p} = p ln M$

Change-of-Base Formula
For any logarithmic bases a and b, and $M > 0,$

$\begin{matrix} {log}_{a} M = \frac{{log}_{b} M}{{log}_{b} a} & {log}_{a} M = \frac{log M}{log a} & {log}_{a} M = \frac{ln M}{ln a} \\ new base b & new base 10 & new base e \end{matrix}$

One-to-One Property of Logarithmic Equations: For $M > 0, N > 0, a > 0,$ and $a \neq 1$ is any real number:

$If {log}_{a} M = {log}_{a} N, then M = N .$
Compound Interest:
For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

$\begin{matrix} A = P {(1 + \frac{r}{n})}^{n t} & when compounded n times a year. \\ A = P e^{r t} & when compounded continuously. \end{matrix}$
Exponential Growth and Decay: For an original amount, $A_{0}$ that grows or decays at a rate, r, for a certain time t, the final amount, A, is $A = A_{0} e^{r t} .$