a. Graph f left parenthesis x right parenthesis equals

Question

6 x minus 9
if x less than or equals 3
x squared
if x greater than 3
b. Find f left parenthesis 1 right parenthesis and f left parenthesis 5 right parenthesis.
c. State the domain of the function.
Question content area bottom
Part 1
a. Choose the correct graph below.
A.
0
6
-10
40
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 6 in increments of 1 and a horizontal y-axis labeled from negative 10 to 40 in increments of 5. The graph consists of a ray and a curve. A ray starts at the closed point (3, 5) and falls from right to left passing through the point (2, negative 1). A curve starts at the open point (3, 9) and rises from left to right at an increasing rate, passing through the point (4, 16).
B.
0
6
-10
40
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 6 in increments of 1 and a horizontal y-axis labeled from negative 10 to 40 in increments of 5. The graph consists of a ray and a curve. A ray starts at the open point (3, 5) and falls from right to left passing through the point (2, negative 1). A curve starts at the closed point (3, 9) and rises from left to right at an increasing rate, passing through the point (4, 16).
C.
0
6
-10
40
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 6 in increments of 1 and a horizontal y-axis labeled from negative 10 to 40 in increments of 5. The graph consists of a ray and a curve. A ray starts at the point (3, 9) and falls from right to left passing through the point (2, 3). A curve starts at (3, 9) and rises from left to right at an increasing rate, passing through the point (4, 16).
D.
0
6
-10
40
x
y

A coordinate system has a horizontal x-axis labeled from 0 to 6 in increments of 1 and a horizontal y-axis labeled from negative 10 to 40 in increments of 5. The graph consists of a ray and a curve. A ray starts at the open point (3, 9) and falls from right to left passing through the point (2, 3). A curve starts at the open point and rises from left to right at an increasing rate, passing through the point (4, 16).
Part 2
b. f left parenthesis 1 right parenthesisequals
  
enter your response here and f left parenthesis 5 right parenthesisequals
  
enter your response here
Part 3
c. Choose the correct answer below.
A.
The domain of the function is xless than or equals3.
B.
The domain of the function is xgreater than3.
C.
The domain of the function is xless than3 and xgreater than3.
D.
The domain of the function is all real numbers.
Choose the correct graph below.b. f left parenthesis 1 right parenthesisf left parenthesis 5 right parenthesisand f left parenthesis 5 right parenthesisChoose the correct answer below.

Answer 1

Let's go through each part of the problem:

Part 1: Graphing the piecewise function

The function \( f(x) \) is defined as:

• \( f(x) = 6x - 9 \) for \( x \leq 3 \)
• \( f(x) = x^2 \) for \( x > 3 \)

1. For \( x \leq 3 \):

   • At \( x = 3 \): \( f(3) = 6(3) - 9 = 18 - 9 = 9 \)
   • At \( x = 2 \): \( f(2) = 6(2) - 9 = 12 - 9 = 3 \)
   • The line will have a slope of 6, starting from the point (3, 9) down to the left.

2. For \( x > 3 \):

   • At \( x = 3 \): \( f(3) \) is already calculated and equals 9. This point is open as the function switches to \( x^2 \) after \( x = 3 \).
   • At \( x = 4 \): \( f(4) = 4^2 = 16 \)

The graph should:

• Include a closed point at (3, 9) and rise on the right side with the curve for \( x^2 \).
• Include an open point at (3, 9) for the linear function's left part.

Choosing the correct graph: Based on the description, the correct answer is likely B (where there is an open point at (3, 9) for the line and a closed point for the quadratic starting at that same vertical height).

Part 2: Evaluating the function.

• For \( f(1) \): Since \( 1 \leq 3 \), we use the first part of the function: \[ f(1) = 6(1) - 9 = 6 - 9 = -3 \]

• For \( f(5) \): Since \( 5 > 3 \), we use the second part of the function: \[ f(5) = 5^2 = 25 \]

Results:

• \( f(1) = -3 \)
• \( f(5) = 25 \)

Part 3: Domain of the function.

The function is defined for all \( x \) (both parts cover the cases for \( x \) less than or equal to 3 and greater than 3). Thus:

• The domain is all real numbers.

So the correct answer is: D. The domain of the function is all real numbers.

Summary:

• a. Choose graph B.
• b. \( f(1) = -3 \) and \( f(5) = 25 \).
• c. The domain of the function is all real numbers (D).