Inverse Functions and Exponential Functions in College Algebra

Inverse Functions

Actions and Their Inverses

Inverse functions "undo" the effect of the original function. For example, if an action is to put on socks and then shoes, the inverse is to take off shoes and then socks. Mathematically, if a function multiplies by 2 and adds 3, its inverse subtracts 3 and then divides by 2.

• Inverse Function: If $f(x)$ is a function, its inverse $f^{-1}(x)$ reverses the effect of $f(x)$.

• Notation: $f^{-1}(x)$ denotes the inverse of $f(x)$.

• Idea: An inverse should "undo" an action.

Definition of Inverse Functions

Two functions $f$ and $g$ are inverses if:

• $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domains of each.

We refer to the inverse function of $f$ as $f^{-1}$.

Numeric Functions and Tables

Numeric tables can be used to evaluate inverse functions. For example:

x	f(x)
-4	3
-3	1
2	4

To find $f^{-1}(4)$, look for the $x$ value where $f(x) = 4$. Here, $f(2) = 4$, so $f^{-1}(4) = 2$.

Graphs of Inverse Functions

• The graph of an inverse function is a reflection of the original function across the line $y = x$.

• Domain and range swap for the inverse function.

• Horizontal asymptotes of the original function become vertical asymptotes for the inverse.

Finding Inverse Functions Algebraically

To find the inverse of a function algebraically:

1. Replace $f(x)$ with $y$.

2. Swap $x$ and $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Example: $f(x) = \sqrt[3]{x+1}$

• Let $y = \sqrt[3]{x+1}$

• Swap: $x = \sqrt[3]{y+1}$

• Cube both sides: $x^3 = y + 1$

• Solve: $y = x^3 - 1$

• So, $f^{-1}(x) = x^3 - 1$

Inverse of Real-World Functions

Given a temperature conversion formula $F = \frac{9}{5}C + 32$, the inverse is:

• Swap $F$ and $C$, solve for $C$:

$F = \frac{9}{5}C + 32$ $F - 32 = \frac{9}{5}C$ $C = \frac{5}{9}(F - 32)$

Self-Inverse Functions

A function is self-inverse if applying it twice returns the original input.

• Example: $g(x) = \frac{6}{x}$ is self-inverse because $g(g(x)) = x$.

• Example: $h(x) = 8 - x$ is self-inverse because $h(h(x)) = x$.

One-to-One Functions and Inverses

A function has an inverse if and only if it is one-to-one (injective).

• Definition: A function is one-to-one if different inputs always result in different outputs.

• Test: No repeated $y$-values for distinct $x$-values.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

Examples of One-to-One and Non-One-to-One Functions

• $f(x) = |x - 7|$ is not one-to-one; no inverse exists.

• $g(x) = \frac{3}{5}x - 8$ is one-to-one; inverse exists.

• $h(x) = -\frac{1}{4}(x+1)^2$ is not one-to-one; no inverse exists.

Exponential Functions and Models

Exponential Functions

An exponential function has the form $f(x) = C a^x$, where $C$ is the initial value and $a$ is the base.

• General Form: $f(x) = C a^x$

• Example: $f(x) = 10(2^x)$, $f(x) = 5(\frac{1}{3})^x$

• Graph passes through $(0, C)$ and increases or decreases depending on $a$.

Exponential Growth and Decay

• Growth: If $a > 1$, the function grows as $x$ increases.

• Decay: If $0 < a < 1$, the function decays as $x$ increases.

• Example: Rabbit population doubles each year: $y = 100 \cdot 2^x$

Compound Interest

Compound interest is calculated using exponential functions. The formula for interest compounded $n$ times per year is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (decimal)

• $n$ = number of times interest is compounded per year

• $t$ = number of years

Example: $P = 1500$, $r = 0.065$, $n = 4$, $t = 5$

$A = 1500 \left(1 + \frac{0.065}{4}\right)^{4 \cdot 5}$

Continuous Compounding and the Number $e$

When interest is compounded continuously, the formula uses the mathematical constant $e \approx 2.71$:

$A = P e^{rt}$

• $e$ is the base of the natural logarithm.

• For continuous compounding, use $A = P e^{rt}$.

Example: To find the principal needed to reach $A = 100,000$ in 18 years at $r = 0.1025$:

$100,000 = P e^{0.1025 \cdot 18}$ $P = \frac{100,000}{e^{0.1025 \cdot 18}}$

Summary Table: Compound Interest Frequencies

Compounding Frequency	n (times/year)
Annually	1
Semi-annually	2
Quarterly	4
Monthly	12
Daily	365

Properties of Exponential Functions

• Exponential functions have constant ratios between outputs for equal steps in $x$.

• The base $a$ determines growth ($a > 1$) or decay ($0 < a < 1$).

• Graphs of exponential functions are always above the $x$-axis for $C > 0$.

Applications

• Population growth

• Compound interest

• Radioactive decay

Additional info: Some explanations and examples have been expanded for clarity and completeness.