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LT 1A – Level 1 (DOK 1 – Recall)
1.5 Analyzing Graphs of Functions
LT 1A – Analyzing Graphs of Functions: I can algebraically and graphically find characteristics of different
functions including: relative minimum/maximum using a graphing calculator, domain and range, intervals of
increase, decrease, constant, positive, negative, symmetries, even and odd functions, end behavior, extending
to modeling situations.
Domain and Range
- Domain: Go from left to right and visualize a vertical line moving across graph and determine the
x-coordinates for which the graph exists. Write domain from least to greatest.
- Range: Go from bottom to top and visualize a horizontal line moving across graph and determine
the y-coordinates for which the graph exists. Write range from least to greatest.
- Notations for Domain and Range:
≥ , ≤, closed circle “●” → use brackets: “[ ]”
, <, open circle “○”, −∞, ∞→ use parenthesis: “( )”
break in the domain and/or range → use the union of “∪”
- Domain of Polynomials: domain is set of all reals.
- Domain of Fractions: domain excludes x-values that result in division by zero (set the factors of the
denominators equal to zero and solve for x; exclude those values).
- Domain of Radicals: domain excludes x-values that result in even roots of negative numbers (set the
factors inside the radical greater than or equal to zero and solve for x).
1 ) Find the domain and range in interval notation.
a) b) c)
Domain: Domain: Domain:
Range: Range: Range:
2 ) Find the domain of each function in interval notation.
a) 𝒇
= √−𝒙 + 𝟐 b) 𝒇
𝟑
− 𝟐 c) 𝒇
𝒙+𝟏
𝒙
𝟐
+𝟑𝒙+𝟐
Evaluating Functions
- Substitute the value inside the parenthesis with every independent variable (usually x). When
substituting for the independent variable, leave the value in the parenthesis in place of the
independent variable. Then simplify.
3 ) Evaluate the function 𝑓(𝑥) = − 2 𝑥
2
- 3 𝑥 + 1 at each specified value of the independent variable.
a) 𝑓(− 2 ) b) 𝑓($)
Increasing, Decreasing, Constant, Positive, Zero, and Negative Functions
- Increasing Function: a function f is increasing on an interval if, for any 𝑥
1
and 𝑥
2
in the interval,
1
2
implies 𝑓(𝑥
1
2
- Decreasing Function: a function f is decreasing on an interval if, for any 𝑥
1
and 𝑥
2
in the interval,
1
2
implies 𝑓(𝑥
1
2
- Constant Function: a function f is constant on an interval if, for any 𝑥
1
and 𝑥
2
in the interval, 𝑓(𝑥
1
2
- Positive Function: A function is positive on intervals (read the intervals on the x-axis), where the graph
line lies above the x-axis.
Negative Function: A function is negative on intervals (read the intervals on the x-axis), where the
graph line lies below the x-axis.
- Zero Function: A function is zero on intervals (read the intervals on the x-axis), where the graph line
lies on the x-axis.
4 ) Determine the intervals over which the function is positive, negative, zero, increasing, decreasing, or
constant.
a) b)
a) intervals over which 𝑓(𝑥) = 0 : a) intervals over which 𝑓(𝑥) = 0 :
b) intervals over which 𝑓(𝑥) > 0 : b) intervals over which 𝑓(𝑥) > 0 :
c) intervals over which 𝑓(𝑥) < 0 : c) intervals over which 𝑓(𝑥) < 0 :
d) intervals over which 𝑓
( 𝑥
) is increasing: d) intervals over which 𝑓
( 𝑥
) is increasing:
e) intervals over which 𝑓
( 𝑥
) is decreasing: e) intervals over which 𝑓
( 𝑥
) is decreasing:
f) intervals over which 𝑓
( 𝑥
) is constant: f) intervals over which 𝑓
( 𝑥
) is constant:
LT 1A – Level 2 (DOK 2 – Routine)
1.5 Analyzing Graphs of Functions
LT 1A – Analyzing Graphs of Functions: I can algebraically and graphically find characteristics of different
functions including: relative minimum/maximum using a graphing calculator, domain and range, intervals of
increase, decrease, constant, positive, negative, symmetries, even and odd functions, end behavior, extending
to modeling situations.
Domain and Range
- Explain why the domain is (−∞, ∞).
Evaluating Function
2 ) Evaluate the function at each specified value of the independent variable.
2
a)𝑓( 2 𝑥) b) 𝑓(𝑥 + ℎ)
Increasing, Decreasing, Constant, Positive, Zero, and Negative Functions
Relative Minimum and Relative Maximum
3 ) a) Graph a function with the following:
x-int: (− 9 , 0 )
Relative Max: (− 4 , 7 )
Relative Mins: all points between
( 0 , 3 ) and ( 6 , 3 )
Increasing: (−∞, − 4 )
Decreasing: (− 4 , 0 )
Constant: ( 0 , 6 )
Increasing: ( 6 , ∞)
b) State the Intervals where the
function is
Positive: 𝑓(𝑥) > 0 :
Zero: 𝑓
Negative: 𝑓(𝑥) < 0 :
- a) State any relative maximums.
b) State any relative minimums.
c) Circle two points are not relative maximums or minimums and explain why.
Even and Odd Functions
- Determine whether the function is even, odd, or neither. Then describe the symmetry.
a) 𝑓
2
3
𝑥
𝑥+ 1
Relative Minimum and Relative Maximum
- Graph a function with the following relative maximum and minimums:
Relative Maximums: (-3, 4), (5, 8)
Relative Minimums: (0, - 5), (8, 2)
Even and Odd Functions
- Give a specific example of an even function and show all steps to prove it is an even function. Then show
why even functions are symmetric with respect to the y-axis.
- Give a specific example of an odd function and show all steps to prove it is an odd function. Then show why
odd functions are symmetric with respect to the origin.
LT 1B – Level 1 (DOK 1 – Recall)
1.6 A Library of Parent Functions
1.7 Transformations of Functions
LT 1B – Transformations of Functions: I can describe and recognize series of transformations applied to the
parent function and then write an equation and graph it, extending to modeling situations. Parent functions
can include constant, linear, absolute value, square root, quadratic, cubic, reciprocal, piecewise, logarithmic
and exponential.
Parent Functions
- State the transformations and graph each function.
a) 𝑓
- Use the description to write a new transformed 𝑔(𝑥) function and graph it.
a) ℎ(𝑥) = 𝑥
2
is translated 3 units left and two units up. b) 𝑓(𝑥) = 𝑥
3
is horizontally stretched by a factor of
Piece-wise Functions
- A function which is defined by multiple sub functions, each sub function applying to a certain interval
of the main function’s domain (a sub-domain).
5 ) Graph the piece-wise function then evaluate the function at each specified value of the independent
variable and simplify.
𝑓(𝑥) = {
−𝑥 + 1 𝑥 < − 2 → 𝑇ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑓(𝑥) = −𝑥 + 1 𝑤ℎ𝑒𝑛 𝑥 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 − 2
2 − 2 ≤ 𝑥 ≤ 2 → 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑓
( 𝑥
) = 2 𝑤ℎ𝑒𝑛 𝑥 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎𝑛𝑑 𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 − 2 𝑎𝑛𝑑 2
𝑥
2
− 5 𝑥 > 2 → 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑓 𝑓
( 𝑥
) = 𝑥
2
− 5 𝑤ℎ𝑒𝑛 𝑥 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 2
a) 𝑓(− 3 ) b) 𝑓(− 2 ) c) 𝑓( 3 )
LT 1B – Level 2 (DOK 2 – Routine)
1.6 A Library of Parent Functions
1.7 Transformations of Functions
LT 1B – Transformations of Functions: I can describe and recognize series of transformations applied to the
parent function and then write an equation and graph it, extending to modeling situations. Parent functions
can include constant, linear, absolute value, square root, quadratic, cubic, reciprocal, piecewise, logarithmic
and exponential.
Parent Functions
- Explain what parent functions are and their role when graphing other functions in the same family.
Transformation of Functions
- State the transformations and graph the function.
- Use the description to write a new transformed function 𝑔
and graph it.
a) 𝑓(𝑥) = √
𝑥 is shifted left 2 units, up 1 unit, b) 𝑔(𝑥) = 𝑥
2
is reflected across the x-axis, shifted
and reflected across the y-axis. down 3 units, and vertically stretched by a factor of
LT 1B – Level 3 (DOK 3 – Non-Routine)
1.6 A Library of Parent Functions
1.7 Transformations of Functions
LT 1B – Transformations of Functions: I can describe and recognize series of transformations applied to the
parent function and then write an equation and graph it, extending to modeling situations. Parent functions
can include constant, linear, absolute value, square root, quadratic, cubic, reciprocal, piecewise, logarithmic
and exponential.
Parent Functions
- Name all the parent functions that comprises each graph and state the graph’s domain and range.
a) Parent Functions b) Parent Functions:
Domain: Range: Domain: Range:
Transformation of Functions
- Explain what happens to the point (-2, 3) if it is reflected over the x-axis. How is this relevant to a function’s
graph that is reflected over the y-axis?
- Use the description to write a new transformed function 𝑔(𝑥) and graph it.
− 3 is stretched vertically by a factor of 3, shifted down
2 units, shifted to the right 4 units, and reflected over the y-axis.
4 ) Write two different functions that represent this transformed graph and explain why these two functions
work by explaining their transformations from the parent function.
Function 1: Function 2:
Transformations Transformations
of Function 1: of Function 1
- Given the graph of 𝑔(𝑥), draw graphs and describe the transformations for a – b.
a) 𝑦 = 𝑔(𝑥) + 2 b) 𝑦 = 𝑔( 3 𝑥)
Piece-wise Functions
6 ) Write a piece-wise function to represent this graph then evaluate the function at each specified value of the
independent variable and simplify.
a) 𝑓(− 7 ) b) 𝑓( 2 ) c) 𝑓( 6 )
g(x)
Inverse Functions
- Inverse Function: 𝑓(𝑥) denoted by 𝑓
− 1
- Steps for Finding Inverse Functions:
- Use the horizontal line test to determine whether or not f has an inverse function.
- Replace 𝑓(𝑥) by y.
- Interchange the roles of x and y.
- Solve for y.
- Replace y by 𝑓
− 1
- Verify that 𝑓(𝑥) and 𝑓
− 1
(𝑥) are inverses of each other either:
a) Graphically: graphs are reflections of each other over the line 𝑦 = 𝑥
b) Algebraically: 𝑓(𝑓
− 1
(𝑥)) = 𝑥 and 𝑓
− 1
- Find the inverse function of the function informally. Verify that 𝑓(𝑓
− 1
) = 𝑥 or 𝑓
− 1
a) 𝑓(𝑥) = 2 𝑥 + 8 b) 𝑓(𝑥) =
𝑥− 1
5
c) 𝑓(𝑥) = (𝑥 − 2 )
3
LT 1C – Level 2 (DOK 2 – Routine)
1.8 Combinations of Functions - Composite Functions
1.9 Inverse Functions
LT 1C – Combinations, Composite and Inverse Functions: I can write both arithmetic combinations and
compositions of functions and determine their domains. I can find an inverse function algebraically and
graphically, restricting the domain of the function when necessary. I can also verify two functions are inverses
by composition.
Operations of Functions
- 𝑓(𝑥) = 𝑥
2
- 2 , 𝑔(𝑥) = 𝑥 − 2 , and ℎ(𝑥) = √
𝑥 + 1. Find each of the following.
a) 𝑓(𝑥 + 1 ) b) (𝑓 − 𝑔)(− 2 ) − 𝑔(𝑥) c) 𝑓 ∘ ℎ d) (
𝑓
𝑔
) (𝑥). Find domain.
𝑥 and 𝑔(𝑥) = √
a) Find (
𝑔
𝑓
) (𝑥). Find domain. b) Find(
𝑓
𝑔
) (𝑥). Find domain.
c) What do you notice about the domains of each when comparing the two.
Composition of Functions
- Find two functions f and g such that 𝑓(𝑔
4
( 5 𝑥+ 2 )
2
LT 1C – Level 3 (DOK 3 – Non-Routine)
1.8 Combinations of Functions - Composite Functions
1.9 Inverse Functions
LT 1C – Combinations, Composite and Inverse Functions: I can write both arithmetic combinations and
compositions of functions and determine their domains. I can find an inverse function algebraically and
graphically, restricting the domain of the function when necessary. I can also verify two functions are inverses
by composition.
Operations of Functions
- The total numbers of Navy personnel N (in thousands) and Marines personnel M (in thousands) from 2000
to 2007 can be approximated by the models
3
2
and 𝑀(𝑡) = 0. 035 𝑡
3
2
where t represent the year, with 𝑡 = 0 corresponding to 2000.
a) Find and interpret (𝑁 + 𝑀)(𝑡). b) Find and interpret (𝑁 − 𝑀)(𝑡).
Evaluate this function for 𝑡 = 6. Evaluate this function for 𝑡 = 6.
Composition of Functions
- You work in the shoe department at Nordstrom and receive a monthly salary. You also receive a 15%
monthly commission on sales over $2,500 for the month. Assume you sell 2,600 this month to get the
commission. Given the functions: 𝑓(𝑥) = 𝑥 − 2 , 500 and𝑔(𝑥) = 0. 15 𝑥, find 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)) when x =
$2,600, then explain in detail which one represents your bonus.
) = Which represents bonus, why:
- You have a coupon for $1 0 0 off an item and another coupon for 15 % off a television at a store that allows
you to use up to two coupons per item.
Let x be the listed price of an item at the store.
a) Write a function D ( x ) representing the cost of b) Write a function P ( x ) representing the cost of
the TV after 15% off. the TV after a $10 0 discount.
c) Find D ( P ( x )) d) Find P ( D ( x ))
e) What is the difference between D ( P ( x )) and P ( D ( x )) according to this problem (besides that they are
different functions/equations)?
f) Which would get you the TV at a lower cost: D ( P ( x )) or P ( D ( x ))? Show your work.
Inverse Functions
4 ) 𝑓(𝑥) = |𝑥 − 4 |
a) Is 𝑓(𝑥) one-to-one? Explain your reasoning. b) Find the inverse graphically. (plot 3 points per graph)
c) Find the inverse algebraically. d) Verify the inverse.
(restrict domain if necessary) (algebraically – one way)
e) 𝐹(𝑥) → Domain: Range:
𝑭
−𝟏
(𝒙) → Domain: Range:
