Key Concept 1

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Key Concept 1. Answer:. Use Set-Builder Notation. A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. The set includes natural numbers greater than or equal to 2 and less than or equal to 7. - PowerPoint PPT Presentation

Transcript of Key Concept 1

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Use Set-Builder Notation

A. Describe {2, 3, 4, 5, 6, 7} using set-builder notation. The set includes natural numbers greater than or equal to 2 and less than or equal to 7.

This is read as the set of all x such that 2 is less than or equal to x and x is less than or equal to 7 and x is an element of the set of natural numbers.

Answer:

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Use Set-Builder Notation

B. Describe x > –17 using set-builder notation.

The set includes all real numbers greater than –17.

Answer:

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Use Set-Builder Notation

C. Describe all multiples of seven using set-builder notation.

The set includes all integers that are multiples of 7.

Answer:

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Describe {6, 7, 8, 9, 10, …} using set-builder notation.

A.

B.

C.

D.

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Use Interval Notation

A. Write –2 ≤ x ≤ 12 using interval notation.

The set includes all real numbers greater than or equal to –2 and less than or equal to 12.

Answer: [–2, 12]

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Use Interval Notation

B. Write x > –4 using interval notation.

The set includes all real numbers greater than –4.

Answer: (–4, )

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Use Interval Notation

C. Write x < 3 or x ≥ 54 using interval notation.

The set includes all real numbers less than 3 and all real numbers greater than or equal to 54.

Answer:

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Write x > 5 or x < –1 using interval notation.

A.

B.

C. (–1, 5)

D.

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Identify Relations that are Functions

A. Determine whether the relation represents y as a function of x.The input value x is the height of a student in inches, and the output value y is the number of books that the student owns.

Answer: No; there is more than one y-value for an x-value.

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Identify Relations that are Functions

B. Determine whether the table represents y as a function of x.

Answer: No; there is more than one y-value for an x-value.

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Identify Relations that are Functions

C. Determine whether the graph represents y as a function of x.

Answer: Yes; there is exactly one y-value for each x-value. Any vertical line will intersect the graph at only one point. Therefore, the graph represents y as a function of x.

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Identify Relations that are Functions

D. Determine whether x = 3y 2 represents y as a

function of x. To determine whether this equation represents y as a function of x, solve the equation for y.

x = 3y 2 Original equation

Divide each side by 3.

Take the square root of each side.

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Identify Relations that are Functions

Answer: No; there is more than one y-value for an x-value.

This equation does not represent y as a function of x because there will be two corresponding y-values, one positive and one negative, for any x-value greater than 0.

Let x = 12.

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Determine whether 12x 2 + 4y = 8 represents y as a

function of x.

A. Yes; there is exactly one y-value for each x-value.

B. No; there is more than one y-value for an x-value.

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Find Function Values

A. If f (x) = x 2 – 2x – 8, find f (3).

To find f (3), replace x with 3 in f (x) = x 2 – 2x – 8.

f (x) = x 2 – 2x – 8 Original function

f (3) = 3 2 – 2(3) – 8 Substitute 3 for x.

= 9 – 6 – 8 Simplify.= –5 Subtract.

Answer: –5

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Find Function Values

B. If f (x) = x 2 – 2x – 8, find f (–3d).

To find f (–3d), replace x with –3d in f (x) = x 2 – 2x – 8.

f (x) = x 2 – 2x – 8 Original function

f (–3d)= (–3d)2 – 2(–3d) – 8 Substitute –3d for x.= 9d

2 + 6d – 8 Simplify.

Answer: 9d 2 + 6d – 8

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Find Function Values

C. If f (x) = x2 – 2x – 8, find f (2a – 1).

To find f (2a – 1), replace x with 2a – 1 in f (x) = x 2 – 2x – 8.

f (x) = x 2 – 2x – 8 Original function

f (2a – 1) = (2a – 1)2 – 2(2a – 1) – 8 Substitute 2a – 1 for x.

= 4a 2 – 4a + 1 – 4a + 2 – 8 Expand

(2a – 1)2 and 2(2a – 1).

= 4a 2 – 8a – 5 Simplify.

Answer: 4a 2 – 8a – 5

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If , find f (6).

A.

B.

C.

D.

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Find Domains Algebraically

A. State the domain of the function .

Because the square root of a negative number cannot

be real, 4x – 1 ≥ 0. Therefore, the domain of g(x) is all

real numbers x such that x ≥ , or .

Answer: all real numbers x such that x ≥ ,

or

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Find Domains Algebraically

B. State the domain of the function .

When the denominator of is zero, the expression

is undefined. Solving t 2 – 1 = 0, the excluded values in

the domain of this function are t = 1 and t = –1. The

domain of this function is all real numbers except

t = 1 and t = –1, or .

Answer:

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Find Domains Algebraically

C. State the domain of the function .

Answer: or

This function is defined only when 2x – 3 > 0.

Therefore, the domain of f (x) is

or .

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State the domain of g (x) = .

A. or [4, ∞)

B. or [–4, 4]

C. or (− , −4]

D.

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A. FINANCE Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage. Their evaluation yielded the following piecewise-defined function. Find the average price per square foot for a home with the square footage of 1400 square feet.

Evaluate a Piecewise-Defined Function

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Evaluate a Piecewise-Defined Function

Because 1400 is between 1000 and 2600,

use to find p(1400).

Subtract.

= 85 Simplify.

Function for 1000 ≤ a < 2600

Substitute 1400 for a.

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Answer: $85 per square foot

Evaluate a Piecewise-Defined Function

According to this model, the average price per square foot for a home with a square footage of 1400 square feet is $85.

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B. FINANCE Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage. Their evaluation yielded the following piecewise-defined function. Find the average price per square foot for a home with the square footage of 3200 square feet.

Evaluate a Piecewise-Defined Function

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Evaluate a Piecewise-Defined Function

Because 3200 is between 2600 and 4000, use

to find p(3200).

Function for

2600 ≤ a < 4000.

Simplify.

Substitute 3200 for a.

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Answer: $104 per square foot

Evaluate a Piecewise-Defined Function

According to this model, the average price per square foot for a home with a square footage of 3200 square feet is $104.

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ENERGY The cost of residential electricity use can be represented by the following piecewise function, where k is the number of kilowatts. Find the cost of electricity for 950 kilowatts.

A. $47.50

B. $48.00

C. $57.50

D. $76.50

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