Question

Pls help me with my math

Answer 1

Answer:

The definition for the given piecewise-defined function is:   $\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}$.

Step-by-step explanation:

General Concepts:

• Piecewise-defined functions.
• Interval notations.

What is a piecewise-defined function?

A piecewise-defined function represents specific rules over different intervals of the domain.  

Symbols used in expressing interval notations:

Open interval: This means that the endpoint is not included in the interval.

We can use the following symbols to indicate the exclusion of endpoints in the interval:

• Left or right parenthesis, "(  )" (or both).

• Greater than (>) or less than (<) symbols.
• Open dot "$\circ$" is another way of expressing the exclusion of an endpoint in the graph of a piecewise-defined function.

Closed interval: This implies the inclusion of endpoints in the interval.

We can use the following symbols to indicate the inclusion of endpoints in the interval:

• Open- or closed brackets (or both), "[  ]."

• Greater than or equal to (≥) or less than or equal to (≤) symbols.
• Closed circle or dot, "•" is another way of expressing the inclusion of the endpoint in the graph of a piecewise-defined function.  

Determine the appropriate function rule that defines different parts of the domain.  

The best way to determine which piecewise-defined function represents the graph is by observing the endpoints and orientation of both partial lines.

• Open circle on (-1, 2):  The graph shows that one of the partial lines has an excluded endpoint of (-1, 2) extending towards the right. This implies that its domain values are defined when x > -1.

• Closed circle on (-1, 1): The graph shows that one of the partial lines has an included endpoint of (-1, 1) extended towards the left. Hence,  its domain values are defined when x ≤ -1.

Based on our observations from the previous step, we can infer that x > -1 or x ≤ -1 apply to piecewise-defined functions A or D. However, only one of those two options represent the graph.

Solution:

a) Test option A:

    $\boxed{\displaystyle\sf Option\:A)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ 2x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ x + 4 & \sf\:{if\:\:x > -1}\end{cases}}$

Piece 1: If x ≤ -1, then it is defined by f(x) = 2x + 2.

We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.

Substitute x = -2 into f(x) = 2x + 2:  

• f(x) = 2x + 2

• f(-2) = 2(-2) + 2
• f(-2) = -4 + 2
• f(-2) = -2  ⇒  False statement.

⇒ The output value of f(-2) = -2 is not included in the graph of the partial line whose endpoint is at (-1, 1).

Piece 2: If x > -1, then it is defined by f(x) = x + 4.

We must choose a domain value that falls within the interval of x > -1 whose output is included in the graph of the partial line with an open dot.

Substitute x = 0 into  f(x) = x + 4:

• f(x) = x + 4
• f(0) = (0) + 4
• f(0) = 4  ⇒  True statement.

⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).

Conclusion for Option A:

Option A is not the correct piecewise-defined function because one of the pieces, f(x) = 2x + 2, does not specify the interval (-∞, -1].

b) Test option D:

    $\boxed{\displaystyle\sf Option\:D)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}$

Piece 1:  If x ≤ -1, then it is defined by f(x) = x + 2.

We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.

Substitute x = -2 into f(x) = x + 2:

• f(x) = x + 2
• f(-2) = (-2) + 2
• f(-2) = 0  ⇒  True statement.

⇒ The output value of f(-2) = 0 is included the graph of the partial line whose endpoint is at (-1, 1).

Piece 2: If x > -1, then it is defined by f(x) = 2x + 4.

We must choose a domain value that falls within the interval of x > -1 whose output is included is included in the graph of the partial line with an open dot.

Substitute x = 0 into f(x) = 2x + 4:

• f(x) = 2x + 4
• f(0) = 2(0) + 4
• f(0) = 0 + 4 = 0  ⇒  True statement.

⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).  

Final Answer:

We can infer that the piecewise-defined function that represents the graph is:

$\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}$.

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Learn more about piecewise-defined functions here:

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