Must show work with some questions
Section 2.2: Circles
KEEP IT 100% MOMENT
It is HIGHLY RECOMMENDED that you either print out
the pdf of this lecture or at least pause the video when
you reach the “TRY FOR YOURSELF” examples and try
to solve them for yourself. It reinforces the brain to
think on its own which reinforces the steps and
processes needed in order to solve the problems. YOU
CANNOT BLAME ANYTHING OR ANYONE FOR LACK OF
EFFORT TO LEARN THE MATERIAL ON YOUR END AND
THAT CAN ONLY COME BY WAY OF PRACTICING.
Center-Radius Form Equation of a Circle.
A circle with center 𝒉, 𝒌 and radius 𝒓 has an equation
𝒙−𝒉
𝟐
+ 𝒚−𝒌
𝟐
= 𝒓
𝟐
Which is the center-radius form of the equation of the circle. A circle
with center 𝟎, 𝟎 and radius 𝒓 has the equation
𝒙 − 𝟎 𝟐 + 𝒚 − 𝟎 𝟐 = 𝒓 𝟐 or 𝒙 𝟐 + 𝒚 𝟐 = 𝒓 𝟐
NOTE: A circle cannot have a negative or imaginary radius!
Center-Radius Form Equation of a Circle.
𝒙−𝒉
𝟐
+ 𝒚−𝒌
𝟐
= 𝒓
𝟐
In CANVAS
𝒙−
𝟐
+ 𝒚−
𝟐
=
Assuming you performed your calculations correctly when you
formed your equation, then all you need to do is enter into the
individual boxes what h, k, and 𝐫 𝟐 are.
NOT Center-Radius Form Equation of a Circle.
𝒙−𝒉 𝟐− 𝒚−𝒌 𝟐 = 𝒓 𝟐
or
𝒙−𝟎 𝟐− 𝒚−𝟎 𝟐 = 𝒓 𝟐
or
𝒙 𝟐− 𝒚 𝟐= 𝒓 𝟐
Finding the Radius of a Circle
If the center 𝒉, 𝒌 is given and one endpoint 𝒙𝟏 , 𝒚𝟏 is known, then
the radius can be found by using the following:
𝒓=
𝒙𝟏 − 𝒉 𝟐 + 𝒚𝟏 − 𝒌 𝟐
If two endpoints 𝒙𝟏 , 𝒚𝟏 and 𝒙𝟐 , 𝒚𝟐 are known, then the radius can be
found by using the following:
𝒓=
𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
𝟒
Finding the Center
If two endpoints 𝒙𝟏 , 𝒚𝟏 and 𝒙𝟐 , 𝒚𝟐 are known, then the center
can be found by using the following:
𝒙𝟏 + 𝒙𝟐
𝒚𝟏 + 𝒚𝟐
𝒉, 𝒌 : 𝒉 =
;𝒌 =
𝟐
𝟐
Ex: Find the center-radius form of the circle.
𝑪𝒆𝒏𝒕𝒆𝒓 −𝟖, −𝟔 , 𝒓𝒂𝒅𝒊𝒖𝒔 𝟏𝟕
𝒉 𝒌
𝑪𝒆𝒏𝒕𝒆𝒓 − 𝑹𝒂𝒅𝒊𝒖𝒔: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓 𝟐
𝒙 − −𝟖
𝟐
+ 𝒚 − −𝟔
𝒙 + 𝟖 𝟐 + 𝒚 + 𝟔 𝟐 = 𝟏𝟕
𝟐
=
𝟏𝟕
𝟐
Ex: Find the center-radius form of the circle.
𝑪𝒆𝒏𝒕𝒆𝒓 𝟓, 𝟖 , 𝒓𝒂𝒅𝒊𝒖𝒔 𝟕
Try for yourself!
Ex: Find the center-radius form of the circle.
𝑪𝒆𝒏𝒕𝒆𝒓 𝟏, 𝟏 , 𝒘𝒊𝒕𝒉 𝒆𝒏𝒅𝒑𝒐𝒊𝒏𝒕 𝟓, 𝟒
𝒙 𝒚
𝒉 𝒌
𝑪𝒆𝒏𝒕𝒆𝒓 − 𝑹𝒂𝒅𝒊𝒖𝒔: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓 𝟐
−𝒉 𝟐+
𝒓=
𝒙𝟏
𝒓=
𝟓−𝟏 𝟐+ 𝟒−𝟏 𝟐
𝒓=
𝟒 𝟐+ 𝟑 𝟐
𝒓 = 𝟐𝟓
𝒓=𝟓
𝒚𝟏
−𝒌 𝟐
𝒙− 𝟏
𝟐
+ 𝒚− 𝟏
𝟐
= 𝟓 𝟐
𝒙 − 𝟏 𝟐 + 𝒚 − 𝟏 𝟐 = 𝟐𝟓
Ex: Find the center-radius form of the circle. Try for yourself!
𝑪𝒆𝒏𝒕𝒆𝒓 −𝟐, 𝟐 , 𝒘𝒊𝒕𝒉 𝒆𝒏𝒅𝒑𝒐𝒊𝒏𝒕 𝟎, 𝟐
Ex: Find the center-radius form of the circle.
𝒂 𝒄𝒊𝒓𝒄𝒍𝒆 𝒘𝒊𝒕𝒉 𝒆𝒏𝒅𝒑𝒐𝒊𝒏𝒕𝒔 −𝟏, 𝟐 𝒂𝒏𝒅 𝟏𝟏, 𝟕
𝒙𝟏 𝒚𝟏
𝒙𝟐 𝒚𝟐
𝑪𝒆𝒏𝒕𝒆𝒓 − 𝑹𝒂𝒅𝒊𝒖𝒔: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓 𝟐
𝒓=
𝒓=
𝒓=
𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
𝟒
𝟏𝟏 − −𝟏
𝟐
+ 𝟕− 𝟐
𝟒
𝟏𝟏 + 𝟏 𝟐 + 𝟕 − 𝟐 𝟐
𝟒
𝒓=
𝟏𝟐 𝟐 + 𝟓 𝟐
𝟒
𝒙𝟏 + 𝒙𝟐
𝒉=
𝟐
𝒚𝟏 + 𝒚𝟐
𝒌=
𝟐
𝒓=
𝟏𝟒𝟒 + 𝟐𝟓
𝟒
−𝟏 + 𝟏𝟏
𝒉=
𝟐
𝟐+𝟕
𝒌=
𝟐
𝒓=
𝟏𝟔𝟗
𝟒
𝟏𝟎
𝒉=
𝟐
𝟗
𝒌=
𝟐
𝟐
𝟏𝟑
𝒓=
𝟐
𝒉=𝟓
Continued→
Ex: Find the center-radius form of the circle.
𝒂 𝒄𝒊𝒓𝒄𝒍𝒆 𝒘𝒊𝒕𝒉 𝒆𝒏𝒅𝒑𝒐𝒊𝒏𝒕𝒔 −𝟏, 𝟐 𝒂𝒏𝒅 𝟏𝟏, 𝟕
𝑪𝒆𝒏𝒕𝒆𝒓 − 𝑹𝒂𝒅𝒊𝒖𝒔: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓 𝟐
𝒉=𝟓
𝟗
𝒌=
𝟐
𝟏𝟑
𝒓=
𝟐
𝟗
𝟐
𝒙− 𝟓
+ 𝒚−
𝟐
𝟗
𝟐
𝒙−𝟓 + 𝒚−
𝟐
𝟐
𝟐
𝟏𝟑
=
𝟐
𝟏𝟔𝟗
=
𝟒
𝟐
Ex: Find the center-radius form of the circle. Try for yourself!
𝒂 𝒄𝒊𝒓𝒄𝒍𝒆 𝒘𝒊𝒕𝒉 𝒆𝒏𝒅𝒑𝒐𝒊𝒏𝒕𝒔 𝟓, 𝟒 𝒂𝒏𝒅 −𝟑, −𝟐
General Form Equation of a Circle.
The equation
𝒙 𝟐 + 𝒚 𝟐 + 𝒄𝒙 + 𝒅𝒚 + 𝒆 = 𝟎
for some real numbers c, d, and e, can have a graph that is a circle or
a point, or is non-existent.
In CANVAS
𝒙^𝟐 + 𝒚^𝟐 + 𝒄𝒙 + 𝒅𝒚 + 𝒆 = 𝟎
Ex: Find the general equation form of a circle.
𝑪𝒆𝒏𝒕𝒆𝒓 −𝟖, −𝟔 , 𝒓𝒂𝒅𝒊𝒖𝒔 𝟑
𝒉 𝒌
𝑪𝒆𝒏𝒕𝒆𝒓 − 𝑹𝒂𝒅𝒊𝒖𝒔: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓 𝟐
𝒙 − −𝟖
𝟐
+ 𝒚 − −𝟔
𝟐
= 𝟑 𝟐
𝒙+𝟖 𝟐+ 𝒚+𝟔 𝟐 =𝟗
𝒙+𝟖 𝒙+𝟖 + 𝒚+𝟔 𝒚+𝟔 =𝟗
𝒙𝟐 + 𝟖𝒙 + 𝟖𝒙 + 𝟔𝟒 + 𝒚𝟐 + 𝟔𝒚 + 𝟔𝒚 + 𝟑𝟔 = 𝟗
𝒙𝟐 + 𝟏𝟔𝒙 + 𝟔𝟒 + 𝒚𝟐 + 𝟏𝟐𝒚 + 𝟑𝟔 = 𝟗
𝒙𝟐 + 𝟏𝟔𝒙 + 𝒚𝟐 + 𝟏𝟐𝒚 + 𝟔𝟒 + 𝟑𝟔 − 𝟗 = 𝟎 → 𝒙𝟐 + 𝒚𝟐 + 𝟏𝟔𝒙 + 𝟏𝟐𝒚 + 𝟗𝟏 = 𝟎
In CANVAS = 𝒙^𝟐 + 𝒚^𝟐 + 𝟏𝟔𝒙 + 𝟏𝟐𝒚 + 𝟗𝟏 = 𝟎
Ex: Find the general equation form of a circle.
𝑪𝒆𝒏𝒕𝒆𝒓 −𝟒, 𝟏 , 𝒓𝒂𝒅𝒊𝒖𝒔 𝟓
Try for yourself!
Ex: Find the center-radius of the circle by completing the square.
𝒙𝟐 + 𝒚𝟐 + 𝟏𝟐𝒙 − 𝟐𝒚 + 𝟐𝟖 = 𝟎
𝒙−𝒉 𝟐+ 𝒚−𝒌 𝟐 = 𝒓 𝟐
𝒙𝟐 + 𝟏𝟐𝒙 + 𝒚𝟐 − 𝟐𝒚 + 𝟐𝟖 = 𝟎
𝑪𝒆𝒏𝒕𝒆𝒓 = 𝒉, 𝒌
𝒙𝟐 + 𝟏𝟐𝒙 + 𝒚𝟐 − 𝟐𝒚 = −𝟐𝟖
𝑹𝒂𝒅𝒊𝒖𝒔 =
𝒓 𝟐
𝒙𝟐 + 𝟏𝟐𝒙 + ? 𝟐 + 𝒚𝟐 − 𝟐𝒚 + ? 𝟐 = −𝟐𝟖 + ? 𝟐 + ? 𝟐
𝒙𝟐 + 𝟏𝟐𝒙 + 𝟔 𝟐 + 𝒚𝟐 − 𝟐𝒚 + −𝟏 𝟐 = −𝟐𝟖 + 𝟔 𝟐 + −𝟏 𝟐
𝒙+𝟔 𝟐+ 𝒚−𝟏 𝟐 =𝟗
𝑪𝒆𝒏𝒕𝒆𝒓 = −𝟔, 𝟏
𝑹𝒂𝒅𝒊𝒖𝒔 = 𝟗 = 𝟑
Ex: Find the center-radius of the circle by completing the square.
𝒙𝟐 + 𝒚𝟐 − 𝟏𝟐𝒙 + 𝟏𝟎𝒚 + 𝟐𝟓 = 𝟎
Try for yourself!
Ex: Find the center-radius of the circle by completing the square.
𝟒𝒙𝟐 + 𝟒𝒚𝟐 + 𝟒𝒙 − 𝟏𝟔𝒚 − 𝟏𝟗 = 𝟎
𝒙−𝒉 𝟐+ 𝒚−𝒌 𝟐 = 𝒓 𝟐
𝟒 𝟐 𝟒 𝟐 𝟒
𝟏𝟔
𝟏𝟗 𝟎
𝒙 + 𝒚 + 𝒙−
𝒚−
=
𝑪𝒆𝒏𝒕𝒆𝒓 = 𝒉, 𝒌
𝟒
𝟒
𝟒
𝟒
𝟒
𝟒
𝟏𝟗
𝟐
𝑹𝒂𝒅𝒊𝒖𝒔
=
𝒓
𝟐
𝟐
𝒙 + 𝒚 + 𝒙 − 𝟒𝒚 −
=𝟎
𝟒
𝟏𝟗
𝟐
𝟐
𝒙 + 𝒙 + 𝒚 − 𝟒𝒚 =
𝟒
𝟏𝟗
𝟐
𝟐
𝟐
𝟐
𝒙 + 𝒙 + ? + 𝒚 − 𝟒𝒚 + ? =
+ ? 𝟐+ ? 𝟐
𝟒
𝟐
𝟐
𝟏
𝟏𝟗
𝟏
𝟐
𝟐
𝟐
𝒙 +𝒙+
+ 𝒚 − 𝟒𝒚 + −𝟐 =
+
+ −𝟐 𝟐
𝟐
𝟒
𝟐
𝟐
𝟏
𝒙+
+ 𝒚−𝟐 𝟐 =𝟗
𝟐
𝟏
𝑹𝒂𝒅𝒊𝒖𝒔 = 𝟗 = 𝟑
𝑪𝒆𝒏𝒕𝒆𝒓 = − , 𝟐
𝟐
Ex: Find the center-radius of the circle by completing the square.
𝟗𝒙𝟐 + 𝟗𝒚𝟐 + 𝟏𝟐𝒙 − 𝟏𝟖𝒚 − 𝟐𝟑 = 𝟎
Try for yourself!
Ex: Graph the circle.
𝒙−𝟏 𝟐+ 𝒚−𝟐 𝟐 =𝟗
𝑪𝒆𝒏𝒕𝒆𝒓 = 𝟏, 𝟐
𝑹𝒂𝒅𝒊𝒖𝒔 = 𝟗 = 𝟑
Ex: Graph the circle.
Try for yourself!
𝒙+𝟐 𝟐+ 𝒚+𝟐 𝟐 =𝟒
Ex: Given the graph of a circle, find the equation of the circle.
Where is the center?
𝑪𝒆𝒏𝒕𝒆𝒓 = −𝟐, 𝟎
What is the radius?
𝑹𝒂𝒅𝒊𝒖𝒔 = 𝟑
Why? The radius is 3 because in each
direction (left, right, up, down) it
takes exactly 3 units to touch the
circle.
𝒙 − −𝟐
𝟐
+ 𝒚− 𝟎
𝒙+𝟐 𝟐+ 𝒚−𝟎 𝟐 =𝟗
𝟐
= 𝟑 𝟐
Ex: Given the graph of a circle, find the equation of the circle.
Try for yourself!
Section 2.3: Functions
KEEP IT 100% MOMENT
It is HIGHLY RECOMMENDED that you either print out
the pdf of this lecture or at least pause the video when
you reach the “TRY FOR YOURSELF” examples and try
to solve them for yourself. It reinforces the brain to
think on its own which reinforces the steps and
processes needed in order to solve the problems. YOU
CANNOT BLAME ANYTHING OR ANYONE FOR LACK OF
EFFORT TO LEARN THE MATERIAL ON YOUR END AND
THAT CAN ONLY COME BY WAY OF PRACTICING.
Section 2.3: Functions
Relation – A relation is a set of ordered pairs.
Function – A relation in which, for each distinct value of the first
component or (x), of the ordered pairs, there is exactly one value
of the second component or (y).
Domain – A domain is the set of all values of the independent
variable or (x) and that value must not repeat. YOU SHOULD LIST
THE VALUES IN ASCENDING ORDER!
Range – A range is the set of all values that are dependent or (y).
YOU SHOULD LIST THE VALUES IN ASCENDING ORDER!
Common Ways to Represent Function Notation.
The following are ways to represent function notation:
𝒇 𝒙 or 𝒈 𝒙 or 𝒉 𝒙 and so on…
Also the most common way to represent a function is by…
𝒚=
Section 2.3: Functions
Ex: Give the domain and range of the relation and determine if the
relation is a function.
𝟑, −𝟏 , 𝟒, 𝟐 , 𝟒, 𝟓 , 𝟔, 𝟖
𝑫𝒐𝒎𝒂𝒊𝒏: 𝟑, 𝟒, 𝟒, 𝟔
𝑹𝒂𝒏𝒈𝒆: −𝟏, 𝟐, 𝟓, 𝟖
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏: 𝑵𝒐! Why? Because there is a value
in the domain that repeats.
Section 2.3: Functions
Ex: Give the domain and range of the relation and determine if the
relation is function. Try for yourself!
𝟐, 𝟑 , 𝟒, 𝟓 , 𝟔, 𝟕 , 𝟖, 𝟗 , 𝟏𝟎, 𝟏𝟏
𝑫𝒐𝒎𝒂𝒊𝒏:
𝑹𝒂𝒏𝒈𝒆:
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
Section 2.3: Functions
Ex: Determine if the relation is function. Give the domain and range of
each relation.
Try for yourself!
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
𝑫𝒐𝒎𝒂𝒊𝒏:
𝑹𝒂𝒏𝒈𝒆:
Section 2.3: Functions
Domain & Range of a Linear Equation of the Forms
𝑨𝒙 + 𝑩𝒚 = 𝑪
𝑫𝒐𝒎𝒂𝒊𝒏: −∞, ∞
𝒚 = 𝒎𝒙 + 𝒃
𝑹𝒂𝒏𝒈𝒆: −∞, ∞
NOTE: It does not matter in which direction
the line maybe going, if the equation is a
linear equation, then the domain & range
follows as previously stated. Linear equations
ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of a Quadratic Equation of the Forms
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
𝒚=𝒂 𝒙−𝒉 𝟐+𝒌
𝑫𝒐𝒎𝒂𝒊𝒏: −∞, ∞
Assuming the graphs
are continuous!
NOTE: The RANGE of a quadratic
𝑹𝒂𝒏𝒈𝒆: −∞, 𝟎
𝑹𝒂𝒏𝒈𝒆: 𝟎, ∞
equation is the interval to which
the values for (y) increase instead
of decrease. Quadratic equations
ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of a Circle of the Form
𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓𝟐
𝑫𝒐𝒎𝒂𝒊𝒏: −𝟑, 𝟑
NOTE: The DOMAIN of a circle equation is the
exact interval to which the circle crosses the
x-axis. The RANGE of a circle equation is the
exact interval to which the circles crosses the
y-axis. Circle equations ARE NOT FUNCTIONS.
𝑹𝒂𝒏𝒈𝒆: −𝟑, 𝟑
Section 2.3: Functions
Domain & Range of a Square Root Equation of the Form
𝒚= 𝒙
𝒇 𝒙 = 𝒙
𝑫𝒐𝒎𝒂𝒊𝒏: ሾ𝟎, ∞ሻ
𝑹𝒂𝒏𝒈𝒆: ሾ𝟎, ∞ሻ
NOTE: The DOMAIN of a square root equation is the
exact interval to which the graph either touches,
crosses, or hovers over or under the x-axis.
Sometimes you may have to solve the square root
equation in order to determine that exactly. The
RANGE of a square root equation is the exact interval
to which the graph crosses or hovers beside the yaxis. Square root equations ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of a Square Root Equation of the Form
𝒚= 𝒙+𝟏
𝒇 𝒙 = 𝒙+𝟏
𝑫𝒐𝒎𝒂𝒊𝒏: ሾ−𝟏, ∞ሻ
𝑹𝒂𝒏𝒈𝒆: ሾ𝟎, ∞ሻ
NOTE: The DOMAIN of a square root equation is the
exact interval to which the graph either touches,
crosses, or hovers over or under the x-axis.
Sometimes you may have to solve the square root
equation in order to determine that exactly. The
RANGE of a square root equation is the exact interval
to which the graph crosses or hovers beside the yaxis. Square root equations ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of a Rational Equation of the Form
𝟏
𝒚=
𝒙
𝑫𝒐𝒎𝒂𝒊𝒏: −∞, 0 ∪ 0, ∞
𝑹𝒂𝒏𝒈𝒆: −∞, 0 ∪ 0, ∞
NOTE: The DOMAIN of a rational equation is the exact
interval to which the graph will not cross the x-axis known
as the vertical asymptote. Sometimes you may have to
solve the rational equation in order to determine that
exactly. The RANGE of a rational equation is the exact
interval to which the graph hovers or runs along the y-axis
known as the horizontal asymptote. Rational equations
ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of an Absolute Value Equation of the Form
𝒚= 𝒙
𝑫𝒐𝒎𝒂𝒊𝒏: ሺ−∞, 𝟎ሿ ∪ ሾ𝟎, ∞ሻ
𝑹𝒂𝒏𝒈𝒆: ሾ𝟎, ∞ሻ
NOTE: The DOMAIN of an absolute value equation is the
exact interval to which the graph will cross, touch, or hover
above or below the x-axis. Sometimes you may have to
solve the absolute value equation in order to determine
that exactly. The RANGE of an absolute equation is the
exact interval to which the graph crosses, hovers or runs
along the y-axis. Absolute Value Equations ARE FUNCTIONS.
Section 2.3: Functions
Domain & Range of an Absolute Value Equation of the Form
𝒚= 𝒙+𝟏
𝑫𝒐𝒎𝒂𝒊𝒏: ሺ−∞, −𝟏ሿ ∪ ሾ−𝟏, ∞ሻ
𝑹𝒂𝒏𝒈𝒆: ሾ𝟎, ∞ሻ
NOTE: The DOMAIN of an absolute value equation is the
exact interval to which the graph will cross, touch, or hover
above or below the x-axis. Sometimes you may have to
solve the absolute value equation in order to determine
that exactly. The RANGE of an absolute equation is the
exact interval to which the graph crosses, hovers or runs
along the y-axis. Absolute Value Equations ARE FUNCTIONS.
Vertical Line Test for Functionality.
The vertical line test is a quick technique used to determine if a
graph is a function. A vertical or straight line is drawn through the
graph and if the lines touches the graph more than once then the
graph is NOT A FUNCTION.
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑵𝒐𝒕 𝒂 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
Vertical Line Test for Functionality.
The vertical line test is a quick technique used to determine if a
graph is a function. A vertical or straight line is drawn through the
graph and if the lines touches the graph more than once then the
graph is NOT A FUNCTION.
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
Section 2.3: Functions
𝑬 𝒙 : Decide whether the equation is a function and then give
the domain and range.
𝒂. 𝒚 = 𝒙 + 𝟒
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏: 𝒀𝒆𝒔! It’s Linear.
𝑫𝒐𝒎𝒂𝒊𝒏: −∞, ∞
𝑹𝒂𝒏𝒈𝒆: −∞, ∞
Section 2.3: Functions
𝑬 𝒙 : Decide whether the equation is a function and then give
the domain and range.
𝒃. 𝒚 = 𝟐𝒙 − 𝟏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏: 𝒀𝒆𝒔!
𝟏
𝑫𝒐𝒎𝒂𝒊𝒏: , ∞ሻ
𝟐
𝟐𝒙 − 𝟏 = 𝟎
+𝟏 +𝟏
𝟐𝒙 = 𝟏
𝟐
𝟐
𝟏
𝒙=
𝟐
𝑹𝒂𝒏𝒈𝒆: ሾ𝟎, ∞ሻ
𝒚 = 𝟐𝒙 − 𝟏
𝒚=
𝟐
𝟏
−𝟏
𝟐
𝒚= 𝟏−𝟏 = 𝟎 =𝟎
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
Section 2.3: Functions
𝑬 𝒙 : Decide whether the equation is a function and then give
the domain and range.
𝟏
𝒄. 𝒚 =
𝒙−𝟏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏: 𝒀𝒆𝒔!
𝑫𝒐𝒎𝒂𝒊𝒏:
𝒙−𝟏=𝟎
+𝟏 +𝟏
𝑹𝒂𝒏𝒈𝒆: −∞, 0 ∪ 0, ∞
𝟏
𝟎=
𝒙−𝟏
𝟎=𝟏
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏
𝒙=𝟏
−∞, 1 ∪ 1, ∞
Notice here that if 𝒚 = 𝟎, we cannot solve
for 𝒙. The 𝒙 would cancel out. Therefore,
𝒚 = 𝟎 cannot be in the range but all other
values for 𝒚 are in the range.
Section 2.3: Functions
𝑬 𝒙 : Decide whether the equation is a function and then give
the domain and range. Try for yourself!
𝒚 = 𝟕 − 𝟐𝒙
𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏:
𝑫𝒐𝒎𝒂𝒊𝒏:
𝑹𝒂𝒏𝒈𝒆:
Section 2.3: Performing
Operations on Functions
Ex: Find 𝒇 −𝒙 when 𝒇 𝒙
𝟐
= −𝟐𝒙 − 𝟑𝒙 − 𝟓
𝒇 −𝒙 = −𝟐𝒙𝟐 − 𝟑𝒙 − 𝟓
𝒇 −𝒙 = −𝟐 −𝒙 𝟐 − 𝟑 −𝒙 − 𝟓
𝒇 −𝒙 = −𝟐𝒙𝟐 + 𝟑𝒙 − 𝟓
Ex: Find 𝒇 𝒘 when 𝒇 𝒙
𝟐
= 𝒙 + 𝟐𝒙 + 𝟖
Try for yourself!
Ex: Find 𝒇 𝒌 − 𝟏 when 𝒇 𝒙
𝟐
= 𝟒𝒙 + 𝟑𝒙 + 𝟒
𝒇 𝒌−𝟏 =𝟒 𝒌−𝟏 𝟐+𝟑 𝒌−𝟏 +𝟒
𝒇 𝒌 − 𝟏 = 𝟒 𝒌𝟐 − 𝒌 − 𝒌 + 𝟏 + 𝟑 𝒌 − 𝟏 + 𝟒
𝒇 𝒌 − 𝟏 = 𝟒 𝒌𝟐 − 𝟐𝒌 + 𝟏 + 𝟑 𝒌 − 𝟏 + 𝟒
𝟐
𝒇 𝒌 − 𝟏 = 𝟒𝒌 − 𝟖𝒌 + 𝟒 + 𝟑𝒌 − 𝟑 + 𝟒
𝟐
𝒇 𝒌 − 𝟏 = 𝟒𝒌 − 𝟓𝒌 + 𝟓
Try for yourself!
𝟐
Ex: Find 𝒇 𝒅 + 𝟐 when 𝒇 𝒙 = 𝒙 − 𝟐𝒙
Ex: Find 𝒈 𝒂 − 𝟏 when 𝒈 𝒙
𝟏
= 𝒙+𝟑
𝟓
𝟏
𝒈 𝒂−𝟏 = 𝒂−𝟏 +𝟑
𝟓
𝟏
𝟏
𝒈 𝒂−𝟏 = 𝒂− +𝟑
𝟓
𝟓
𝟏
𝟏𝟒
𝟏
𝟏 𝟏𝟓
= 𝒂+
𝒈 𝒂−𝟏 = 𝒂− +
𝟓
𝟓
𝟓
𝟓 𝟓
Try for yourself!
Ex: Find 𝒉 𝒛 + 𝟐 when 𝒉 𝒙
𝟐
𝟏
=− 𝒙+
𝟑
𝟐
Determining if a graph is defined or not along the y-axis
for a given or known value of x.
1) If for a given value x there is a hole in the graph along the
y-axis, then x is not defined or does not exist (DNE) at that
location.
2) If for a given value x there is not a hole in the graph along
the y-axis, then x is defined at that location.
Ex: Find 𝒇 𝟎 , 𝒇 𝟒 and 𝒇 −𝟏 if they exist.
𝒇 𝟎 = 𝟒
𝒇 𝟒 = 𝟒
𝒇 −𝟏 = 𝟐
Ex: Find 𝑯 𝟎 , 𝑯 𝟑 and 𝑯 −𝟐 if they exist.
𝑯 𝟎 =𝟑
𝑯 𝟑
Does Not Exist
or
=
DNE
𝑯 −𝟐 = 𝟐
Ex: Find 𝒇 𝟎 , 𝒇 −𝟐 and 𝒇 −𝟏 if they exist.
Try for yourself!
Section 2.3: Increasing,
Decreasing, Constant Functions
Increasing, Decreasing, Constant Functions.
1) 𝒇 increases on an interval when 𝒙𝟏 < 𝒙𝟐 & 𝑓 𝒙𝟏 < 𝑓 𝒙𝟐
2) 𝒇 decreases on an interval when 𝒙𝟏 < 𝒙𝟐 & 𝑓 𝒙𝟏 > 𝑓 𝒙𝟐
3) 𝒇 constant on an interval when 𝒙𝟏 𝑎𝑛𝑑 𝒙𝟐 & 𝑓 𝒙𝟏 = 𝑓 𝒙𝟐
For graphing using open interval notation purposes, parentheses ( )
are used.
For graphing using closed interval notation purposes, brackets [ ]
are used.
Ex: Find increasing, decreasing, and constant intervals
of the function below if they exist.
Increasing, Decreasing, Constant Functions.
1) 𝒇 increases on an interval when 𝒙𝟏 < 𝒙𝟐 & 𝑓 𝒙𝟏 < 𝑓 𝒙𝟐
2) 𝒇 decreases on an interval when 𝒙𝟏 < 𝒙𝟐 & 𝑓 𝒙𝟏 > 𝑓 𝒙𝟐
3) 𝒇 constant on an interval when 𝒙𝟏 𝑎𝑛𝑑 𝒙𝟐 & 𝑓 𝒙𝟏 = 𝑓 𝒙𝟐
𝒇 𝒙 = 𝒙𝟑 − 𝟏 on the interval (-4,4).
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: −𝟒, 𝟒
𝒇 −𝟒 = −𝟒 𝟑 − 𝟏 𝒇 𝟒 = 𝟒 𝟑 − 𝟏
𝒇 𝟒 = 𝟔𝟒 − 𝟏
𝒇 −𝟒 = −𝟔𝟒 − 𝟏
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: 𝑵𝒐𝒏𝒆
𝒇 −𝟒 = −𝟔𝟓
𝒇 𝟒 = 𝟔𝟑
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕: 𝑵𝒐𝒏𝒆
Ex: Find increasing, decreasing, and constant intervals
of the function below if they exist.
𝒇 𝒙 = − 𝒙 − 𝟒 on the interval (4,7).
Try for yourself!
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕:
Ex: Find increasing, decreasing, and constant intervals
of the graph if they exist.
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: ሾ𝟏, ∞ሻ
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: ሺ−∞, −𝟏ሿ
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕:
−𝟏, 𝟏
Ex: Find increasing, decreasing, and constant intervals
of the graph if they exist.
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: 𝑵𝒐𝒏𝒆
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈: ሺ−∞, 𝟎ሿ ∪ ሾ𝟎, ∞ሻ
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕:
𝑵𝒐𝒏𝒆
Ex: Find increasing, decreasing, and constant intervals
of the graph if they exist.
Try for yourself!
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕:
Ex: Find increasing, decreasing, and constant intervals
of the graph if they exist.
Try for yourself!
𝑰𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑫𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈:
𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕:
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