# Math 3C Review Sheet

## 1 Functions in general

1. Ways of expressing: table, graph, formula, words

2. Input and output; domain and range. Interval notation.

3. Describing change of a function: average rate of change, increasing and
decreasing functions, concavity

4. Short run behavior: zeros, vertical asymptotes, holes, etc.

5. Long run behavior: horizontal asymptotes, dominance, long-run similarity of
functions

6. Periodic functions: period, amplitude, midline. Trigonometric functions as
examples. See below.

7. Piecewise defined functions. Example: absolute value function f(x) = |x|.

8. Operations on functions

(a) Composition f(g(x))

(b) Arithmetic combinations f(x) + g(x), f(x)g(x) etc.

(c) Inverse function f^{-1}(y): find input for a given output

i. Horizontal line test for invertible functions

ii. Finding formula by solving y = f(x) for x

iii. Graph: reflection about diagonal line y = x, domain and range swap

iv. Composition of inverses: f(f^{-1}(x)) = f^{-1}(f(x)) = x

v. Examples: see below.

(d) Transformations of functions

i. Inside and outside changes; horizontal and vertical

ii. Shifts, stretches/compressions, reflections/flips.

iii. Even and odd functions

## 2 Specific families of functions

1. Linear functions

(a) Constant rate of change (slope). Graph: straight line.

(b) Formulas for linear functions: slope-intercept form y = mx+b, pointslope

form y − y_{0} = m(x - x_{0}), standard form Ax + By + C = 0.

(c) Solving linear equations and linear systems. Parallel and perpendicular
lines.

2. Exponential and logarithmic functions

(a) Properties of exponents and logarithms: log(ab) = log a + log b, etc.

(b) Graphs and general shape. Example of inverse functions

(c) Solving equations using/involving exponents and logarithms. Finding formulas
for exponential functions.

(d) Applications: interest, population growth, radioactive decay, logarithmic
scales, etc.

3. Trigonometric functions: sin t, cos t, tan t

(a) Definitions in terms of unit circle. Special angles
etc. Symmetry properties, even and odd functions.

(b) Radian measure and arc length

(c) Sinusoidal functions: transformations of sin t and cos t. Effect on period,
amplitude, midline, phase shift.

(d) Trigonometric identities: Pythagorean identity, double angle formula, phase
shifts by

(e) Inverse trigonometric functions: arcsin t or sin^{-1} t, etc. Restriction of
domain to avoid failure of horizontal line test.

(f) Applications

i. Solving right triangles: adjacent/opposite/hypotenuse

ii. Solving non-right triangles: law of sines, law of cosines

iii. Modeling periodic behavior

4. Power functions f(x) = kx^{p}.

(a) Shape of graph for various values of p

(b) Negative p: vertical and horizontal asymptotes

(c) Even and odd functions

(d) Finding formulas using logarithms

5. Quadratic functions: f(x) = ax^2 + bx + c

(a) Graph: parabola. Concave up or down.

(b) Finding zeros: factoring, quadratic formula

(c) Finding vertex: vertex form (transformation of g(x) = x^2), completing the
square.

6. Polynomials

(a) Sum of power functions. Standard form, leading term, degree.

(b) Long run behavior: leading term

(c) Short run behavior: zeros and factoring

(d) Formula <-> graph

7. Rational functions
where p(x), q(x) are polynomials

(a) Long run behavior: divide leading terms of p(x) and q(x).

(b) Short run behavior: find zeros of p(x), q(x) by factoring. They correspond

to zeros, vertical asymptotes, and holes for r(x).

(c) Formula <-> graph

8. Dominance: f(x) > g(x) when x is large

(a) Exponential functions: compare growth factor

(b) Power functions: compare power

(c) Increasing functions: exponential dominates power dominates logarithmic

(d) Decreasing functions: power dominates exponential, i.e. power functions
decrease slower.