Math quiz on functions

Input

x-value

Output

y-value

Function Notation

• A notation that describes a function

• For the function F, when x is the input, f(x) is the output

Domain and Range

• Domain = All possible inputs (x-values)

• Range = All possible outputs (y-values)

Interval Notation

• Uses brackets and an interval to show the set of all numbers between two values

  • (__,__) or (__,__] or [__,__]

Set Notation

• Specifies a set of elements that satisfy a set of conditions

  • {xIx<__}

In set notation, what does the variable and then a line mean (xI or yI)

It means its for domain or range

Hard bracket vs. Soft bracket

• Hard bracket [ = A set point

• Soft bracket ( = Everything up to the number but not the number or numbers that aren’t real (like infinity)

x-intercept

• Also called zeroes, roots, and solutions

• Value of domain where f(x) = 0

• Where the line intercepts the x-axis

• Any number of x-intercepts in a function

y-intercept

• Value of range when f(0) = y

• When the line intercepts the y-axis

• At most one y-intercept in a function

Asymptote

• A line that a curve approaches as one or both of the variables in the equation of the curve approach infinity

• Curve never touches the line

End Behavior

• Trend of a function or graph at its left and right ends; specifically, each y-value that each end, obtains or approaches

• Ex. As x —> ∞, f(x) —> -∞

What does the arrow in the end behavior statement mean

It means “approaches”

What does it mean if the end of a function is a line or arrow going off the page?

It means it goes on forever

Absolute maximum and Absolute minimum

• Absolute maximum = Highest y-value across the domain of a function

• Absolute minimum = Lowest y-value across the domain of a function

• A function can have at most one absolute, maximum or minimum, but that value might occur at multiple different x-coordinates

Relative maximum and Relative minimum

• A point that is higher than all the neighboring points immediately around it (top of a “hill”)

• A point that is lower than all the neighboring points immediately around it (bottom of a “valley”)

What do the symbols in this set notation equation mean?

• x in the beginning represents “set of all x’s”

• E like symbol means “that are members of

• R like symbol means “real numbers”

Function Families…

…have a parent function and all transformations of the parent function

In this example: ∛, what is the index?

The 3

What are all of the function families?

• Square root function

• Cube root function

• Rational function (both linear and quadratic denominators

• Exponential

• Logarithmic function

• Absolute value function

Square root function

• Radical function with an index of 2

• Parent function → f(x) = √x, where x ≥ 0

• Domain = [ 0, ∞ )

• Range = [ 0, ∞ )

• Absolute minimum at origin / No absolute maximum

• Increasing over its domain

What does “increasing over its domain” mean?

As you move from left to right, or as the x-value gets bigger, the y-value gets bigger too.

Cube root function

• Radical function with an index of 3

• Parent function → f(x) = ∛x

• Domain = ( -∞, ∞ )

• Range = ( -∞, ∞ )

• No relative or absolute maximum or minimum

• x and y intercept at origin

• Increasing over its domain

Rational function

• Quotient of a polynomial with a denominator of at least 1

• Two rational parent functions: One with a linear denominator, and one with a quadratic denominator

Rational parent function with linear denominator

• Parent function → f(x) = 1/x

• Domain = ( -∞, 0) U ( 0, ∞ )

• Range = ( -∞, 0) U ( 0, ∞ )

• No absolute or relative maximum or minimum

• No x or y intercepts

• Asymptotes at the lines x=0 and y=0

• Decreasing over its domain

Which is horizontal and which is vertical: x=0 and y=0

• x=0 is vertical

• y=0 is horizontal

What do the domain and range of the rational parent function with linear denominator include?

They include all real numbers except for 0

What doed the domain of the rational parent function with quadratic denominator include?

It includes all real numbers except for 0

Rational parent function with quadratic denominator

• Parent function → f(x) = 1/x²

• Domain = ( -∞, 0) U ( 0, ∞ )

• Range = ( 0, ∞ )

• No absolute or relative maximum or minimum

• No x or y intercepts

• Asymptotes at the lines x=0 and y=0

• Increasing over ( -∞, 0 ) ← interval

• Decreasing over ( 0, ∞ ) ← interval

What does “decreasing over its domain mean”?

As x-values go left or right, or get bigger, the y-values consistently go down

Exponential function

• Independent variable (x) is in the exponent

• Parent function → f(x) = ab^x where a ≠ 0 and b > 0

• Domain = ( -∞, ∞ )

• Range = ( 0, ∞ )

• Horizontal asymptote at y=0

• Increasing when b > 1

• Decreasing when 0 < b < 1

• No absolute or relative maximum or minimum

• No x-intercept, y-intercept at origin

Logarithmic function

• Represents the exponent to which b must be raised to get x

• Parent function → f(x) = log_bx where x > 0, b > 0, and b ≠ 1

• Domain = ( 0, ∞ )

• Range = ( -∞, ∞ )

• Increasing when b > 1

• Decreasing when 0 < b < 1 (fraction)

• No absolute or relative maximum or minimum

• x-intercept at (1,0), no y-intercept

• Asymptote at the vertical x=0

What is “b”?

• b = base

• Controls the rate of growth and decay

Absolute value function

• Contains a variable expression inside absolute value bars

• Parent function → f(x) = a Ix-hI + k

• Domain = ( -∞, ∞ )

• Range = [ 0, ∞ )

• Absolute minimum at origin

• x and y intercepts at origin

• Vertical line of symmetry seperating the increasing and decreasing intervals

Full template for transformations

g(x) = a · f [ b (x-h) ] + k

In the template for transformations, what does a represent?

Vertical dilations

In the template for transformations, what does (x-h) represent?

Horizontal shift

In the template for transformations, what does + k represent?

Vertical shift

What are the two types of translations?

Horizontal and vertical shifts (translations)

Vertical Shift

• Represented by k

• At the end of the equation, just simple plus or minus for up and down

Horizontal Shift

• Represented by (x-h)

• A negative number means shift to the right and vice versa

What are the two types of dilations?

Horizontal and vertical dilations

Vertical dilations

• Represented by I a I

• g(x) = I a I · f(x)

• Compression = 0 < IaI < 1 (fractions)

• Stretch = IaI > 1

Horizontal dilations

• Represented by I b I

• g(x) = f ( IbI · x )

• Compression = I b I > 1

• Stretch = 0 < I b I < 1 (fractions)

What are the subcategories of horizontal and vertical dilations and what do they mean?

• Horizontal

  • Compression = towards x-axis

  • Stretch = away from x-axis

• Vertical

  • Compression = towards y-axis

  • Stretch = away from y-axis

Reflections

• Across x-axis → g(x) = -f(x)

• Across y-axis → g(x) = f(-x)

What function is this?

Square root function

What function is this?

Cube root function

What function is this?

Rational function with linear denominator

What function is this?

Rational function with quadratic denominator

What function is this?

Logarithmic function

What function is this?

Exponential

What function is this?

Absolute value function

What is the vertex in a formula?

( h, k )