Lesson on Functions and Graphs

Functions & Graphs

​[Self Learning mode]

by FathiahJasni

​There are 4 arrow diagram relations. But which of those are actually functions?

We can use vertical line test to determine a function:

​If the vertical line intersect the graph at one point, then the graph is a function.

Multiple Choice

Which type of relation is the following mapping diagram?

• one-to-one 

• one-to-many 

• many-to-one

• many-to-many

Multiple Choice

Is relation shown a function?

• YES

• NO

Multiple Choice

Is this a function? Explain.

• Yes; it passes the horizontal line test.

• No; it is not a straight line.

• Yes; it passes the vertical line test.

• No; it does not pass the vertical line test

Multiple Choice

Is y a function of x?

• function

• not a function

Multiple Choice

Is y a function of x?

• function

• not a function

There are actually only 2 types of relations that contribute to function. ​

Multiple Select

What are the relations that are actually functions?

Tick those who are applicable.

• one-to-one

• one-to-many

• many-to-one

• many-to-many

Multiple Select

How do you determine that a function is one-to-one?

Tick all that applicable

• Vertical Line Test

• Horizontal Line Test

• Algebraically find the relation from  $$f\left(x_1\right)=f\left(x_2\right)$$  

• Check whether it is ONTO

Multiple Choice

Use the horizontal line test to determine if the function is one-to-one.

• yes

• no

Multiple Choice

Use the horizontal line test to determine if the function is one-to-one.

• yes

• no

Multiple Choice

Use the horizontal line test to determine if the function is one-to-one.

• yes

• no

​Let's try the algebraic approach method.

​Well, that was pretty easy, right?

Multiple Select

By using algebraic approach, which of the following statements describe $$f\left(x\right)=3x-9$$  ?

• It is not one-to-one

• It is one-to-one

• $$x_1=x_2$$  

• $$x_1=3x_2$$  

Multiple Select

By using algebraic approach, which of the following statements describe  $$g\left(x\right)=x^2-9,\ x\ge-1$$  ?

• It is one-to-one

• It is not one-to-one

• $$x_1=x_2$$  

• $$x_1=\pm x_2$$  

Multiple Select

By using algebraic approach, which of the following statements describe  $$h\left(x\right)=1-x^2,\ x\ge2$$  ?

• It is one-to-one

• It is not one-to-one

• $$x_1=x_2$$  

• $$x_1=\pm x_2$$  

Did you notice that you've learned 10 types of functions in this topic?

​Do you think you can spot the correct shapes, find their domains and ranges?

Multiple Choice

What is the RANGE?

• {-3, 3]

• all real numbers

• [-4, 5]

• [-4, ∞)

Multiple Choice

What is the equation of the asymptote for this graph?

• x = 1

• y = 1

• x = 0

• y = 0

Multiple Choice

What is the domain of this function? $$y=\sqrt{x-1}-3$$  

• {x ∣ x ≥ 1}

• {x ∣ x ≤ 1}

• {x ∣ x ≥ -3}

• {x ∣ x ≥ -1}

Multiple Choice

Identify the Range

• (- ∞, -2 ]

• [ -2, ∞ )

• All Real Numbers

• [ 0, ∞)

Multiple Choice

What movie are these characters from?

• Hercules 

• Beauty & the Beast

• Tangled 

• Frozen

Multiple Choice

What is the domain for the following function?

• All real numbers except 2.

• All real number except -2.

• All real numbers except 4.

• All real numbers.

Multiple Choice

Which of the following is the function shown in the graph?

Multiple Choice

What is the domain of the function?

• x > 2

• x > 3

• All Real Numbers

Multiple Choice

What is the y-intercept of the function?

• 2

• 3

• 1

• -2

Multiple Select

select all the cubic graphs similar to the cubic parent function....

Multiple Choice

Find the corresponding graph of $$-\log_2\left(x-3\right)$$  

​You're half way to go with the functions!

Come on!​

Multiple Choice

What is the DOMAIN of the function?

• [-3, infinity)

•  [0, infinity)

• [3, inifinity)

• (-infinity, 0}

Multiple Choice

Which of the following could be a graph of $$y=3x^3$$ ?

Multiple Choice

Which of the following represents the graph below?

Multiple Choice

What is this function called?

• Quadratic

• Reciprocal

• Polynomial

• Exponential

Multiple Choice

What is the horizontal asymptote of the following function?

• x = -2

• x = 2

• y = -3

• y = 3

Multiple Choice

Which function describes the graph?

• $$y=-\sqrt{x+1}$$

• $$y=\sqrt{x+1}$$

• $$y=-e^x$$

• $$y=-e^{-x}$$

Multiple Choice

How many Harry Potter movies are there?

• 5

• 9

• 8

• 7

Multiple Choice

What is the range of the function graphed?

• (-∞, ∞)

• (-∞, 2] and [2, ∞)

• [2, ∞)

• (-∞, 2]

Multiple Choice

What is the horizontal asymptote of the function?

• x =1

• x = -2

• x = 2

• y = 0

Multiple Choice

What type of function is this?

• Linear

• Exponential

• Cubic

• Constant

Multiple Choice

What is the domain in Interval notation?

• [ -2 , 2 ]

• [ 4 , -4 ]

• -2 ≤ x ≤ 2

• 4 ≥ y ≥ -4

Multiple Choice

What is the domain of f(x) = 5^x?

• (-∞ ,∞)

• (-∞ ,0)

• (0,∞)

Multiple Choice

Which is the range of the function shown?

• $$x<-3\ \cup\ x>-3$$

• $$y<-3\ \cup\ y>-3$$

• $$x<4\ \cup\ x>4$$

• $$y<4\ \cup\ y>4$$

Multiple Choice

Which of the following has a vertical asymptote at $$x=4$$  

• $$y=2^x-4$$  

• $$y=\log_2\left(x-4\right)$$  

• $$y=2^x+4$$  

• $$y=\log_2\left(x+4\right)$$  

Multiple Choice

The range in the equation y= x³ is .....

• $$\left(-\infty,\infty\right)$$

• $$\left[-\infty,\infty\right]$$

• $$\left(\infty,-\infty\right)$$

• $$\left(-\infty,\infty\right]$$

Multiple Choice

Find the corresponding graph of $$\log_3\left(x-2\right)$$  

​There are 4 basic operations on functions.

• ​Addition

• ​Subtraction

• ​Multiplication

• ​Division

​If we have 2 function f(x) and g(x), then:

​The not-so basic operation on functions is the composite function.

​For Example,

Multiple Choice

Evaluate $$\left(\frac{g}{h}\right)\left(x\right)$$   , given: g(x)= 2x, h(x) = 5x

• 23

• 2/5

• 5/2

• 10

Multiple Choice

If $$f\left(x\right)\ =\ x-1$$ and $$g\left(x\right)=5x-2$$ , then  $$\left(f\cdot g\right)\left(x\right)=$$

• $$5x-3$$  

• $$5x^2+2$$  

• $$5x^2-7x+2$$  

• $$5x-7$$  

Multiple Choice

Find f(3) + g(2) if f(x)= x+2 and g(x)=5x-1

• 2
• 5
• 9
• 14

Multiple Choice

If  $$f\left(x\right)=x-6$$  and  $$g\left(x\right)=\sqrt[]{x-5}$$  , What is the domain for  $$f\left(x\right)-g\left(x\right)$$  ?

• $$\left(-\infty,\infty\right)$$  

• $$\left(-\infty,6\right)$$  

• $$\left[6,\infty\right)$$  

• $$\left[5,\infty\right)$$  

Multiple Choice

f(x) = 3x + 10
g(x) = x - 2
Find f(g(5))

• 19
• 23
• -10
• None of these.

Multiple Choice

Given that, $$w\left(x\right)=2x+3$$   and $$v\left(x\right)=\frac{5}{x}$$  . Find $$w\ \circ\ v$$  .

• $$\frac{10+3x}{x}$$  

• $$\frac{5}{2x+3}$$  

• $$10x+\frac{3}{x}$$  

• $$4x^2+9$$  

Multiple Choice

If  $$f\ \circ\ g\ \left(x\right)=x^2+1$$  , find  $$f\left(x\right)$$   when  $$g\left(x\right)=x+3$$  

• $$\frac{-x^2}{2}$$  

• $$x^2-6x+10$$  

• $$x^2-x-2$$  

• $$x^2+4$$  

​You're the best!