Chapter 7.1-7.2: Functions and Graphs

Functions and Graphs

Chapter 7.1-7.2

What is a function?

• A function is a correspondence (relationship) between a first set and a second set

• The first set (potential inputs) is called the domain, often the x-coordinate

• The second set (potential outputs) is called the range, often the y-coordinate (or f(x))

• A correspondence is a function if one input has exactly one output

• In other words, one input (or x) -> one output (or y)

Multiple Choice

Is the following correspondence a function: {(-1,  $$\frac{1}{2}$$  ), (0, 1), (1, 2), (2, 4), (3, 8)}

• yes

• no

Yes, this is a function because each input (x-coordinate) has exactly one output (y-coordinate).

Elementary and Intermediate Algebra | Ch7

Multiple Choice

What is the domain of the function:

$$\left\{(-1,\ \frac{1}{2}),\ (0,\ 1),\ (1,\ 2),\ (2,\ 4),\ (3,\ 8)\right\}$$  

• { $$\frac{1}{2}$$ , 1, 2, 4, 8}

• [-1, 3]

• {-1, 0, 1, 2, 3}

• $$\left[\frac{1}{2},\ 8\right]$$

The domain is the set of all inputs, each x-coordinate.

{-1, 0, 1, 2, 3}

Elementary and Intermediate Algebra | Ch7

Multiple Choice

What is the range of the function:

$$\left\{(-1,\ \frac{1}{2}),\ (0,\ 1),\ (1,\ 2),\ (2,\ 4),\ (3,\ 8)\right\}$$  

• { $$\frac{1}{2}$$ , 1, 2, 4, 8}

• [-1, 3]

• {-1, 0, 1, 2, 3}

• $$\left[\frac{1}{2},\ 8\right]$$

Elementary and Intermediate Algebra | Ch7

Find the member of the range that is paired with -2.

OR

Find f(-2)​

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Finding output given input using a graph

Find the member of the domain that is paired with 6.

OR

Find x if f(x)=6​

​

Finding input given output using a graph

Multiple Choice

For the function g, determine g(2).

• $$g\left(2\right)=1$$  

• $$g\left(2\right)=0$$  

• $$g\left(2\right)=3.5$$  

• $$g\left(2\right)=-3$$  

Multiple Choice

For the function g, determine any x-values for which $$g\left(x\right)=-3$$  .

• $$x=2\ \text{or}\ g\left(2\right)=-3$$  

• $$x=-5\ \text{or}\ g\left(-5\right)=-3$$  

The Vertical-Line Test

• If it is possible for a vertical line to cross a graph more than once...

• Then the graph is not the graph of a function

Dropdown

The graph to the left

Select answer

a function.

It is not possible for a vertical line to cross the graph more than once.

​

The graph to the left is a function.

Dropdown

The graph to the left

Select answer

a function.

It is possible for a vertical line to cross the graph more than once.

The graph to the left is not a function.

Function Notation and Equations

Function Notation and Equations

Example 1

Example 2

Multiple Choice

Find f(-1) for $$f\left(x\right)=2-3x$$

• f(-1) = -1

• f(-1) = 1

• f(-1) = 5

• f(-1) = 6

Multiple Choice

Find f(2a) for $$f\left(x\right)=\ x^2+2x-3$$

• $$f\left(2a\right)=4a^2+4a-3$$

• $$f\left(2a\right)=2a^2+2a-3$$

• $$f\left(2a\right)=-3$$

• $$f\left(2a\right)=1$$

Multiple Choice

Find f(a+1) for $$f\left(x\right)=3x+2$$

• $$f\left(a+1\right)=3a+5$$

• $$f\left(a+1\right)=3a+6$$

• $$f\left(a+1\right)=3a+3$$

• $$f\left(a+1\right)=6$$

Multiple Choice

Find $$g\left(a\right)+2$$ given the function $$g\left(x\right)=2x+5$$

• $$g\left(a\right)+2=9+a$$

• $$g\left(a\right)+2=7$$

• $$g\left(a\right)+2=2a+9$$

• $$g\left(a\right)+2\ =\ 2a+7$$

Domain of a Function

Multiple Choice

What is the domain of the function to the left?

• $$\left(-\infty,\ \infty\right)$$

• $$\text{[}-3,\ \infty\text{)}$$

• $$\text{[0, }\infty\text{)}$$

This is a linear function. The domain of linear functions is all real numbers.

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Multiple Choice

Domain and range

• Domain: [-3, 1];

  Range: [-4, 4]

• Domain: (-4, 4]; Range: (-3, 1)

• Domain: [-4, 4];

  Range: [-3, 1]

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

  $$\text{Range:}\left(-\infty,\ \infty\right)$$

Multiple Choice

Determine the Domain and Range.

• $$\text{Domain:}\ ℝ$$ $$\text{Range:}\ ℝ$$

• $$\text{Domain: }\left\{x\text{|}x\ge-3\right\}$$

  $$\text{Range}\text{:}\ ℝ$$

• $$\text{D:}\ \left\{x\text{|}x=ℝ,\ x\ne3\right\}$$ $$\text{R:}\left\{f\left(x\right)\text{|}\ f\left(x\right)=ℝ,\ f\left(x\right)\ne0\right\}$$

• $$\text{Domain:}\ \left\{x\text{|}x\ge-3\right\}$$ $$\text{Range: }\left\{f\left(x\right)\text{|}f\left(x\right)\ge0\right\}$$

Multiple Choice

Find the domain and range of the function.

• Domain: $$\left[-5,\ 4\right]$$

  Range: $$\left[-5,\ 3\right]$$  

• Domain $$\left\{x\text{|}-5\le x\le4\right\}$$  

  Range: $$\left\{y\text{|}-5\le y\le3\right\}$$  

• Domain: $$\left\{-5,\ 1,\ 3,\ 4\right\}$$  

  Range: $$\left\{-5,\ 0,\ 1,\ 3\right\}$$  

Multiple Choice

Determine the domain of the following function: $$f\left(x\right)=\frac{x+2}{x-7}$$  

• $$ℝ$$  

• $$\left\{x\text{|}x=ℝ,\ x\ne-2,\ x\ne7\right\}$$  

• $$\left\{x\text{|}x=ℝ,\ x\ne7\right\}$$  

• $$x\ \text{is a real number and }x\ne7$$  

Multiple Choice

• $$\left(-∞,\ ∞\right)$$  

• $$\left\{p\text{|}p\ge5.5\right\}\ \text{or}\ \left[5.5,\ ∞\right)$$  

• $$\left\{p\text{|}p\ge-5.5\right\}\ \text{or}\ \left[-5.5,\ ∞\right)$$  

• $$\left\{p\text{|}0\le p\le5.5\right\}\ \text{or}\ \left[0,\ 5.5\right]$$  

Multiple Choice

Find $$f\left(6\right)$$  

• $$f\left(6\right)=69$$  

• $$f\left(6\right)=36$$  

• $$f\left(6\right)=23$$  

Multiple Choice

Find $$f\left(0\right)$$  

• $$f\left(0\right)=-3$$  

• $$f\left(0\right)=0$$  

• $$f\left(0\right)=-7$$  

Multiple Choice

Find $$f\left(3\right)$$  

• $$f\left(3\right)=15$$  

• $$f\left(3\right)=9$$  

• $$f\left(3\right)=8$$