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Domain and Range of a Function

Some text here about the topic of discussion

The range of a function is the set of all possible output values (y)​ for the function.​ Includes all of the numbers on the vertical number line.

Range

The domain of a function is the set of all possible input values (x)​ for the function. Includes all the numbers on the horizontal number line

Domain

Absolute Value Function is written as f(x)= |x|. The graph looks like a V; the turning point is called the vertex.

PARENT FUNCTION

Graphing Absolute Value Functions with Transformations

Parameters a, h, k

a- slope

h-left or right

k-up or down

Parameters of the Absolute Value Function:

• a Compresses or stretches the graph vertically (Graph will Push TO and Pull AWAY from the X-axis)

• h Translates the graph of the absolute value function either to the left (+h) or right (-h) !

• k Translates the graph of the absolute value function either to the up (+k) or down (-k) !

Multiple Select

Select all that apply. (May have more than one correct answer)

f(x) = |x - 6| + 2

• Horizontal translation right 6 units

• Horizontal translation left 6 units

• Vertical translation up 2 units

• Vertical translation down 2 units

Steps to Graphing

• Find the vertex (-h, k)

• Graph the vertex

• EITHER ...

• A. Move according to the slope

• B. Create a table to represent your function and graph these points

Multiple Choice

Choose the correct equation for the graph.

• $$y=\left|x-6\right|-3$$

• $$y=\left|x-6\right|+3$$

• $$y=\left|x+6\right|+3$$

• $$y=\left|x+6\right|-3$$

Multiple Choice

Choose the correct equation for the graph.

•  $$y=\left|x+2\right|$$

• $$y=\left|x-2\right|$$

• $$y=\left|x\right|+2$$

• $$y=\left|x\right|-2$$

Multiple Choice

Choose the correct equation for the graph.

•  $$y=\left|x+2\right|$$

• $$y=\left|x-2\right|$$

• $$y=\left|x\right|+2$$

• $$y=\left|x\right|-2$$

​Horizontal Translation

This allows our function to move LEFT or RIGHT

​

​** These are inside the parentheses **

​

(x – h) shifts right

(x + h) shifts left

​Vertical Translation

​This allows our function to move UP or DOWN

​

​** These are outside of the parentheses **

​​

​+ k shifts up​

– k shifts down​

Reflection

When there is a negative sign in front of the equation, this means the function is being reflected over the x-axis!​

Vertical Stretch and Vertical Shrink​

Makes the graph more narrow or more wide

​

​a > 1 = Vertical Stretch

0 < a < 1 = Vertical Shrink

Multiple Choice

Name the type of function:

• piecewise function

• linear function

• quadratic function

• step function

Multiple Choice

Given the piecewise function, evaluate $$f\left(2\right)$$  

• 2

• 6

• -2

• 14

Multiple Choice

Which type of function does the graph represent?

• Horizontal Function

• Linear Function

• Step Function

• Exponential Function

Multiple Choice

How much does it cost to rent the Karaoke machine for 3 days?

• $125

• $100

• $150

• $75

Multiple Choice

Morgan can start wrestling at age 5 in Division 1. He remains in that division until his next odd birthday when he is required to move up to the next division level. Which graph correctly represents this information?

Multiple Choice

Evaluate  $$f\left(4\right)$$  

• 2

• 4

• 6

• 0

Multiple Choice

What are the domain restrictions for the green piece of this function?

• $$-2<x\le1$$

• $$-1<x\le0.5$$

• $$x\le-2$$

• $$x>-2$$  

Multiple Choice

What is the domain of the absolute value function?

• x > 0

• x > 4

• All Real Numbers

• x < 0

Multiple Choice

What is the range of the absolute value function?

• y > 0

• y > 4

• All Real Numbers

• y < 4

Limits

We will use the graph of f(x) shown to the right to answer some questions about limits.

Multiple Choice

Using the graph of f(x), what is

$$\lim_{x\rightarrow2^-}f\left(x\right)$$  ?

• $$\lim_{x\rightarrow2^-}f\left(x\right)=-2$$  

• $$\lim_{x\rightarrow2^-}f\left(x\right)=-1$$  

• $$\lim_{x\rightarrow2^-}f\left(x\right)=0$$  

• DNE

Multiple Choice

Use the graph of  $$f\left(x\right)$$   to find

$$\lim_{x\rightarrow2^+}f\left(x\right)$$  .

• $$\lim_{x\rightarrow2^+}f\left(x\right)\ =\ -2$$  

• $$\lim_{x\rightarrow2^+}f\left(x\right)=-1$$  

• $$\lim_{x\rightarrow2^+}f\left(x\right)=0$$  

• DNE

Multiple Choice

Use the graph of  $$f\left(x\right)$$   to find

$$\lim_{x\rightarrow2^{ }}f\left(x\right)$$  .

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)\ =\ -2$$  

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)=-1$$  

• $$\lim_{x\rightarrow2^{ }}f\left(x\right)=0$$  

• DNE

Multiple Choice

Use the graph of  $$f\left(x\right)$$   to find

$$\lim_{x\rightarrow4^-}f\left(x\right)$$  .

• $$\lim_{x\rightarrow4^-}f\left(x\right)\ =\ 0$$  

• $$\lim_{x\rightarrow4^-}f\left(x\right)=1$$  

• $$\lim_{x\rightarrow4^-}f\left(x\right)=2$$  

• DNE

Multiple Choice

Use the graph of  $$f\left(x\right)$$   to find

$$\lim_{x\rightarrow4^+}f\left(x\right)$$  .

• $$\lim_{x\rightarrow4^+}f\left(x\right)\ =\ 0$$  

• $$\lim_{x\rightarrow4^+}f\left(x\right)=1$$  

• $$\lim_{x\rightarrow4^+}f\left(x\right)=2$$  

• DNE

​To understand what limits are, let's look at an example.

We start with the function f(x)=x+2

​

​

The limit of f at x=3 is the value f approaches as we get closer and closer to x=3. Graphically, this is the y-value we approach when we look at the graph of f and get closer and closer to the point on the graph where x=3.​

​For example, if we start at the point (1,3) and move on the graph until we get really close to x=3 then our y-value (i.e. the function's value) gets really close to 5.

​Similarly, if we start at (5,7) and move to the left until we get really close to x=3 y-value again will be really close to 5.

​For these reasons we say that the limit of f at x=3 is 5.

​

​You might be asking yourselves what's the difference between the limit of f at x=3 and the value of f at x=3, i.e. f(3).

​

​So yes, the limit of f(x)=x+2 at x=3 is equal to f(3), but this isn't always the case. To understand this, let's look at function g. This function is the same as f in every way except that it's undefined at x=3.

​Just like f, the limit of g at x=3 is 5. That's because we can still get very very close to x=3 and the function's values will get very very close to 5.

​

So the limit of ggg at x=3 is equal to 5, but the value of g at x=3 is undefined! They are not the same!

​

That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.

​

Multiple Choice

What is a reasonable estimate for the limit of h at x=3?

• 2

• 3

• 4

• The limit does not exist!

​We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3

The symbol "lim" means we're taking a limit of something.

The expression to the right of "lim" is the expression we're taking the limit of. In our case, that's the function f.

The expression x→3 that comes below "Iim" means that we take the limit of f as values of x approach 3.

Multiple Choice

What is a reasonable estimate for $$\lim_{x\to6\ }f\left(x\right)$$  

• -5

• -3

• 6

• The limit does not exist!

Multiple Choice

Which expression represents the limit of $$x^2$$  as x approaches 5?

• $$\lim\ 5^2$$  

• $$\lim_{x^2\to\ 5}$$  

• $$\lim_{x\to5}x^2$$  

• $$\lim_{x\to25}x$$  

A limit must be the same from both sides.

​

Now take, for example, function hhh. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.

When we approach x=3 from the left, the function approaches 4. When we approach x=3 from the right, the function approaches 6.

​

When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.

​

Multiple Select

Which of the limit exist?

• $$\lim_{x\to3}g\left(x\right)$$  

• $$\lim_{x\to5}g\left(x\right)$$  

• $$\lim_{x\to6}g\left(x\right)$$  

• $$\lim_{x\to7}g\left(x\right)$$  

Multiple Choice

Find $$\lim_{x\rightarrow0-}f\left(x\right)$$  .

• 0

• -5

• -6

• Does Not Exist

Multiple Choice

Find $$\lim_{x\rightarrow0+}f\left(x\right)$$  .

• 0

• -5

• -6

• Does Not Exist

Multiple Choice

Find $$\lim_{x\rightarrow0}f\left(x\right)$$  .

• 0

• -5

• -6

• Does Not Exist

Multiple Choice

Find $$\lim_{x\rightarrow5}f\left(x\right)$$  .

• 0

• -5

• -6

• Does Not Exist

​We have special notation to talk about limits.....

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​If a limit Does Not Exist write DNE or dne

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Definition of a Limit

Limits from a Graph

* Limits at a point

* Left & Right Limits

* Limit Does No Exist