Domain, Range, and Characteristics of Logarithms

Characteristics of Exponential and Logarithmic Functions

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Vocabulary

• Asymptote: A line that a function approaches

• Domain: The set of all x-values. (Look left and right)

• Range: The set of all y-values (Look down and up)

PLEASE TAKE NOTES!

PLEASE TAKE NOTES!

Multiple Choice

A given exponential function has an asymptote at y=12. What must be the asymptote of the inverse?

• $$y=\frac{1}{12}$$

• $$x=12$$

• $$x=-12$$

• $$y=-12$$

Multiple Choice

A given logarithmic function has an asymptote at x=5. What must be the asymptote of the inverse?

• 
  

  $$y=\frac{1}{12}$$  

• $$x=12$$  

• $$x=5$$

• $$y=5$$

Multiple Choice

A given exponential function has an asymptote at y=-1. What must be the asymptote of the inverse?

• $$y=-1$$

• $$x=-1$$

• $$x=1$$

• $$y=1$$

PLEASE TAKE NOTES!

Multiple Choice

An exponential function has a domain of:

$$\left(-\infty,\infty\right)$$  
What is the range of the inverse function?

• $$\left(-\infty,\infty\right)$$  

• $$\left(\infty,-\infty\right)$$  

• Cannot determine from the given information.

Multiple Choice

An exponential function has a range of:

$$\left(3,\infty\right)$$  
What is the domain of the inverse function?

• $$\left(\infty,3\right)$$  

• $$\left(3,\infty\right)$$  

• Cannot determine from the given information.

Multiple Choice

An logarithmic function has a domain of:

$$\left(-1,\infty\right)$$  
What is the domain of the inverse function?

• $$\left(\infty,1\right)$$  

• $$\left(-1,\infty\right)$$  

• Cannot determine from the given information.

PLEASE TAKE NOTES!

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PLEASE TAKE NOTES!

PLEASE Predict what the answer is...

Multiple Choice

What is the asymptote of the yellow and green function.

• yellow: y=0

  green: x=0

• yellow: y=1

  green: x=1

• yellow: y=-1

  green: x=-1

• yellow: y=-1

  green: x=1

Multiple Select

What is the domain and range of the green function? (Choose 2)

• Domain:

  $$\left(-1,\infty\right)$$

• Domain:
  $$\left(-\infty,\infty\right)$$
  

• Range: $$\left(-1,\infty\right)$$

• Range: $$\left(-\infty,\infty\right)$$

Multiple Select

What is the domain and range of the Yellow function? (Choose 2)

• Domain:

  $$\left(-1,\infty\right)$$

• Domain:
  $$\left(-\infty,\infty\right)$$
  

• Range: $$\left(-1,\infty\right)$$

• Range: $$\left(-\infty,\infty\right)$$

Notice how our domain and range coincide with end behavior?

Open Ended

How can end behavior help us find the domain and range of the function? Don't worry, there is no wrong answer here.

Multiple Choice

Choose the correct choice for end behavior.

• 
  

  $$as\ \left(x\rightarrow-\infty\right),\ \left(y\rightarrow0\right);\ as\ \left(x\rightarrow\infty\right),\left(\ y\rightarrow\infty\right)$$
  

• $$as\ \left[x\rightarrow0\right],\ \left[y\rightarrow-\infty\right];\ as\ \left[x\rightarrow\infty\right],\ \left[y\rightarrow\infty\right]$$

• $$as\ \left[x\rightarrow0,\ y\rightarrow0\right];\ and\ as\ \left[x\rightarrow\infty,\ y\rightarrow\infty\right]$$

Multiple Choice

Logarithmic functions are the inverse of...

• Linear Functions
• Exponential Functions 
• Quadratic Functions 
• Polynomial Functions 

Multiple Choice

Which graph best represents the following logarithmic function?
$$y=\log_4x$$  

Multiple Choice

Which graph best represents the following logarithmic function?
$$y=\log_2x$$  

Multiple Choice

Which function has a horizontal asymptote?

• exponential

• logarithm

• both

• neither

Multiple Choice

Which function has a vertical asymptote?

• exponential

• logarithm

• both

• neither

Multiple Choice

Classify the following graph.

• Exponential Growth

• Exponential Decay

• Logarithmic

• Rational

Multiple Choice

Where is the asymptote?

• x = 1
• y = 1
• x = -1
• y = -1

Multiple Choice

What is the domain?

• (-∞, ∞)
• (0, ∞)
• [1, ∞)
• (1, ∞)

Multiple Choice

What is the range?

• (-∞, ∞)
• (0, ∞)
• [1, ∞)
• (1, ∞)

Multiple Choice

What is the base?

• 1
• 5
• 10
• 3

Multiple Choice

What is the domain of log(x)?

• (-∞, ∞)

• [0, ∞)

• (0, ∞)

• (∞, -∞)

Multiple Choice

What is the range of log(x)?

• (-2, 4)

• (-∞, ∞)

• (-∞, 0)

• (∞, -∞)

Multiple Choice

What is the x-intercept of log₇(x)?

• (2, 0)

• (-1, 0)

• (0, 1)

• (1, 0)

Multiple Choice

The exponential function f(x)=2^x and the logarithmic function g(x)=log₂(x) are inverses. You know that because their graphs are reflected over what line?

• the x-axis

• the y-axis

• y=x

• y=-x

Multiple Choice

Find the corresponding graph of
$$y=-\log_4\left(x+1\right)+2$$  

Multiple Choice

Find the corresponding graph of
$$y=\log_2\left(-x\right)-1$$  

The end!
But is it? How about some Disney Halloween Trivia?
Let's play shall we.... *Cue evil laugh...

Multiple Choice

Disney collaborated with the creators of The Twilight Zone to create a short story for a ride. What is that ride?

• Tower of Terror

• Tower of Doom

• Hollywood Tower

• Twilight Tower

Multiple Choice

Ichabod Crane is chased through Sleepy Hollow by which villain?

• The Caped Horseman

• The Headless Horseman

• The Devil's Horseman

• The Four Horsemen

Multiple Choice

Who is the main character/hero in the film The Nightmare Before Christmas?

• Jack Sparrow

• Jack Skellington

• Jack Pumpkin

• Jack Skeleton

Multiple Choice

Which attraction at Walt Disney World features a stretching room with paintings of various characters' deadly demises?

• The Haunted Mansion

• The Haunted Manor

• Phantom Manor

• The Haunted Mickey

Multiple Choice

What is the name of the song played continuously throughout Mickey's Not So Scary Halloween Party parade?

• Let's Boo This

• Boo To You

• Let's Get Tricky

• It's Halloween Everybody

Multiple Choice

Who is this?

• Cruella Deville

• Ursula

• Tiana

• Vanessa

Multiple Choice

What does she give Snow White?

• A poison apple

• A cookie

• A pie

• A banana