#1 - What You Should Memorize Before Unit #1 Exam

f₀g

f( g(x) )

Input g(x) into f(x)

g₀f

g ( f(x) )

Input f(x) into g(x)

Domain for Radicals in the Denominator

Radicals should be set > 0

Ex. g(x) =  √(9-x²) / √(x+5) 

9 - x² must be ≥ 0 and x + 5 must be > 0.

9 - x² ≥ 0; x + 5 > 0

Domain for Radicals in the Numerator, or Radicals not in a fraction

Radicals should be set ≥ 0 

Ex. h(x) = √(4-x) + √(x²-1)

4 - x ≥ 0; x²-1 ≥ 0

Domain for Polynomials

Ex. linear, quadratic, cubic, …

• No restrictions.

• Domain: (−∞,∞), aka ALL REAL numbers

Domain for Rational Functions

• Denominator ≠ 0.

• Exclude any x that makes the denominator 0.

• Example: 1/(x-2) → Domain: (-∞, 2) ∪ (2, ∞)

Domain for Even Root Radicals

Even-Root Radicals (like square root, 4th root)

• Inside the radical ≥ 0 (must be nonnegative).

• If in numerator or standalone → ≥ 0

• If in denominator → > 0

• Example:√(x+3) = → x + 3 ≥ 0 → Domain:  [-3, ∞)

Domain for Odd Root Radicals

Odd-Root Radicals (cube root, 5th root, …)

• No restriction — all real numbers work. 

Example in image.

Domain for Absolute Value Functions

(-∞, ∞) 

All real numbers

Domain of Combined Functions

Where both f(x) and g(x) are defined—domain is intersection of all VALID intervals

Domains of Quotients (f/g) Functions

Given (f/g) = ( f(x) ) / ( g(x) ),

The domain is everything for f(x) and g(x), but g(x) cannot equal 0.

x = A ∩ B | g(x) ≠ 0

Vertifical Shifting of Functions

f(x) ± c

f(x) + c → shift the function UP by "c" units

f(x) - c → shift the function DOWN by "c" units

Horizontal Shifting of Functions

f(x±c)

f(x + c) → shift the function LEFT by "c" units

f(x - c) → shift the function RIGHT by "c" units

THINK: The Opposite of What You'd Suspect, if you didn't know function transformations

Reflecting Graphs on the X-Axis

-f(x) → reflect the graph across the X-AXIS

THINK: outside x, inside y--in regards to the negative

Reflecting Graphs on the Y-axis

f(-x) → reflect the graph across the Y-AXIS

THINK: outside x, inside y--in regards to the negative

-f(x)

Reflect the graph across the X-AXIS

THINK: outside x, inside y--in regards to the negative

f(-x)

Reflect the graph across the Y-AXIS

THINK: outside x, inside y--in regards to the negative

y = cf(x)

Vertical Stretching or Shrinking, multiply the y-coordinates by "c"

c > 1 → STRETCH

0 < c < 1 → SHRINK

y = cf(x), c > 1

VERTICAL STRETCH, as c is not a fraction and is greater than 1

y = cf(x), 0 < c < 1

VERTICAL SHRINK, as c is greater than zero and less than 1 (aka a fraction)

y = f(cx)

Horizontal Stretching or Shrinking, multiply the x-coordinates by "c"

c > 1 → STRETCH

0 < c < 1 → SHRINK

y = f(cx), c > 1

HORIZTONAL STRETCH, as c is greater than 1 and is not a fraction

y = f(cx), 0 < c < 1

HORIZONTAL SHRINK, as c is less than 1 but greater than zero, being a fraction

Even or Odd: f(-x) = f(x)

EVEN

Describe the graph of an EVEN function

The graph is symmetrical, with respect to its y-axis

Even or Odd: f(-x) = -f(x)

ODD

Describe the graph of an ODD function

The graph is symmetrical, with respect to the origin

Identify how to determine if a function is one-to-one, using a graph.

You can apply the horizontal line test.

Horizontal Line Test

A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Identify what is true about One-to-One Functions

1) Passes Horizontal Line Test

2) Every increasing and decreasing function is considered one-to-one

3) No 2 inputs have the same output

Inverse Function Property

If both provided functions are the inverse of each other and are plugged into one another, they should simplify to x = x on both sides.

f^-1( f(x) ) = x

f( f^-1(x) ) = x

Domain of Inverse Functions

You switch the original function x and y to get the domain of an inverse function, as x and y switch.

|x| < c

-c < x < c

|x| ≤ c

- c ≤ x ≤ c

|x| > c

x < -c or x > c

|x| ≥ c

x ≤ -c or x ≥ c