Review

Review algebraic transformations and their conditions, function and graph relationships, composition and inverses, geometric series, recurrence, and convergence.

Algebra changes the form of a relationship while preserving its meaning. Functions describe how outputs depend on inputs, and sequences describe values indexed in an order. This review collects the definitions, conditions, and checks used throughout the chapter.

Comparing Quantities

Fractions, ratios, and rates all use division, but they answer different questions.

Idea	Meaning	Example
Fraction	A number written as one quantity divided by another; context may make it a part-to-whole comparison	$3/8$ of the observations
Ratio	One quantity compared with another	$3:5$ positive to negative cases
Rate	One quantity per unit of another	$60$ kilometres per hour
Proportion	An equality between two ratios	$a/b=c/d$

A weighted average divides a weighted sum by the total weight:

\bar{x}_w=\frac{\sum_i w_i x_i}{\sum_i w_i}, \qquad \sum_i w_i\ne0.

The denominator is part of the meaning. Changing it can turn a part-to-whole fraction into a different comparison.

Equations and Valid Transformations

An equation asks which values make two expressions equal. Adding or subtracting the same quantity on both sides preserves its solutions. Multiplying or dividing both sides by the same nonzero quantity also preserves them.

Restrictions must be recorded before simplifying. For example:

\frac{x^2-1}{x-1}=x+1

is valid only when $x\ne1$ , because the original denominator is zero at $x=1$ . A simplified expression may hide a restriction; it does not remove it.

Expansion and factoring move in opposite directions:

(a+b)^2=a^2+2ab+b^2,

x^2-9=(x-3)(x+3).

Substitution into the original equation is a useful final check, especially after squaring, taking roots, or dividing by an expression that contains a variable.

Powers, Roots, Exponentials, and Logarithms

For a nonzero base $a$ and integer exponents $m$ and $n$ :

a^m a^n=a^{m+n}, \qquad \frac{a^m}{a^n}=a^{m-n}, \qquad (a^m)^n=a^{mn}.

Remember the difference between a principal root and all solutions of an equation:

\sqrt{9}=3, \qquad x^2=9 \Longrightarrow x=\pm3.

Over the real numbers:

\sqrt{x^2}=|x|.

An exponential function $b^x$ requires $b>0$ and $b\ne1$ when it is paired with a logarithm. The logarithm answers which exponent produces a positive number:

y=b^x \quad\Longleftrightarrow\quad x=\log_b y, \qquad y>0.

For positive $u$ and $v$ :

\log_b(uv)=\log_b u+\log_b v, \qquad \log_b\!\left(\frac{u}{v}\right)=\log_b u-\log_b v.

A logarithm of zero or a negative real number is not defined in the real number system.

Inequalities and Distance

An inequality describes a set of allowed values. Multiplying or dividing by a negative number reverses its direction:

-2x<6 \quad\Longrightarrow\quad x>-3.

Absolute value measures distance on the number line. For $r\ge0$ :

|x-c|\le r \quad\Longleftrightarrow\quad c-r\le x\le c+r,

while

|x-c|\ge r \quad\Longleftrightarrow\quad x\le c-r \text{ or } x\ge c+r.

Closed interval endpoints are included; open endpoints are excluded. Thus $[a,b)$ includes $a$ but excludes $b$ .

Functions and Graphs

A function assigns exactly one output to each input in its domain. Its graph contains the points $(x,f(x))$ . Read a graph by identifying its domain, range, intercepts, zeros, increasing and decreasing regions, bounds, symmetry, and long-run behavior.

Common transformations of $y=f(x)$ include:

Rule	Effect on the graph
$f(x)+k$	Shift vertically by $k$
$f(x-h)$	Shift horizontally by $h$
$af(x)$	Scale outputs by $a$ ; reflect vertically when $a$ is negative
$f(bx)$ , $b\ne0$	Scale inputs by $1/

The sign inside a function acts in the opposite horizontal direction from what a first glance may suggest. Check a known point when uncertain.

Composition and Inverses

Composition passes one function's output into another:

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

Apply the inner function first, and require its output to lie in the domain of the outer function. In general, $g\circ f$ and $f\circ g$ are different.

An inverse reverses a function. On the relevant domains:

f^{-1}(f(x))=x, \qquad f(f^{-1}(y))=y.

An inverse function exists only when every output identifies one input. A domain restriction can sometimes make a non-injective rule invertible. The notation $f^{-1}$ means inverse function, not reciprocal; the reciprocal is $1/f(x)$ .

Sequences, Series, and Convergence

An explicit sequence rule computes a term from its index. A recurrence uses earlier terms and therefore also needs enough initial conditions.

For an arithmetic sequence with difference $d$ :

a_n=a_0+nd.

For a geometric sequence with ratio $r$ :

b_n=b_0r^n.

The finite geometric sum is:

1+r+r^2+\cdots+r^n =\frac{1-r^{n+1}}{1-r}, \qquad r\ne1.

If $|r|<1$ , its partial sums converge and:

1+r+r^2+\cdots=\frac{1}{1-r}.

A sequence converges to $L$ when its terms eventually remain within every positive tolerance of $L$ :

\lim_{n\to\infty}a_n=L.

For a recurrence $x_{n+1}=F(x_n)$ , a possible limit must often satisfy the fixed-point equation $F(L)=L$ . Finding a fixed point identifies a candidate; it does not by itself prove convergence.

Checks That Prevent Common Errors

Before accepting a result, ask:

What does each quantity represent, and what units does it have?
Which values are excluded by denominators, roots, or logarithms?
Did every algebraic step preserve the original solution set?
If an inequality was multiplied or divided, was the sign of that quantity known?
Does the function input lie in its domain?
Was a composition evaluated from the inside outward?
Does an alleged inverse recover inputs in both directions?
Does a recurrence include its initial conditions?
Is a long-run pattern merely suggested by examples, or justified as a convergence claim?

These habits carry directly into vector algebra. The next chapter replaces single numbers with ordered collections while keeping the same attention to meaning, valid operations, shape, and scale.