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a. Graph f left parenthesis x right parenthesis equals

4 x minus 4
if x less than or equals 2
x squared
if x greater than 2
b. Find f left parenthesis 1 right parenthesis and f left parenthesis 4 right parenthesis.
c. State the domain of the function.
Question content area bottom
Part 1
a. Choose the correct graph below.
A.
-1
5
-10
30
x
y

A coordinate system has a horizontal x-axis labeled from negative 1 to 5 in increments of 1 and a horizontal y-axis labeled from negative 10 to 30 in increments of 5. The graph consists of a ray and a curve. A ray starts at the closed point (2, 0) and falls from right to left passing through the point (1, negative 4). A curve starts at the open point (2, 4) and rises from left to right at an increasing rate, passing through the point (4, 16).
B.
-1
5
-10
30
x
y

A coordinate system has a horizontal x-axis labeled from negative 1 to 5 in increments of 1 and a horizontal y-axis labeled from negative 10 to 30 in increments of 5. The graph consists of a ray and a curve. A ray starts at the open point (2, 0) and falls from right to left passing through the point (1, negative 4). A curve starts at the closed point (2, 4) and rises from left to right at an increasing rate, passing through the point (4, 16).
C.
-1
5
-10
30
x
y

A coordinate system has a horizontal x-axis labeled from negative 1 to 5 in increments of 1 and a horizontal y-axis labeled from negative 10 to 30 in increments of 5. The graph consists of a ray and a curve. A ray starts at the open point (2, 4) and falls from right to left passing through the point (1, 0). A curve starts at the open point and rises from left to right at an increasing rate, passing through the point (4, 16).
D.
-1
5
-10
30
x
y

A coordinate system has a horizontal x-axis labeled from negative 1 to 5 in increments of 1 and a horizontal y-axis labeled from negative 10 to 30 in increments of 5. The graph consists of a ray and a curve. A ray starts at the point (2, 4) and falls from right to left passing through the point (1, 0). A curve starts at (2, 4) 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 4 right parenthesisequals
  
enter your response here
Part 3
c. Choose the correct answer below.
A.
The domain of the function is xless than or equals2.
B.
The domain of the function is xgreater than2.
C.
The domain of the function is xless than2 and xgreater than2.
D.
The domain of the function is all real numbers.

1 answer

Let's break this down step by step.

Part 1: Graph of the Function

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

  • \( f(x) = 4x - 4 \) if \( x \leq 2 \)
  • \( f(x) = x^2 \) if \( x > 2 \)

To analyze it:

  1. For \( x \leq 2 \):

    • \( f(2) = 4(2) - 4 = 8 - 4 = 4 \)
    • The linear function \( f(x) = 4x - 4 \) extends from point \( (2, 4) \) to the left, and it is a downward-sloping ray.

  2. For \( x > 2 \):

    • The function is quadratic (specifically \( f(x) = x^2 \)), which starts at an open point \( (2, 4) \) and opens upwards.

Now, we can look for the graph that correctly represents this description:

  • The graph must have a downward ray starting at \( (2, 4) \) and covering values less than or equal to \( 2 \).
  • The quadratic part must start from \( (2, 4) \) and curve upwards from there.

Option D is the correct answer, as it shows:

  • A ray starting from \( (2, 4) \) and falling to the left.
  • An upward-opening curve starting from \( (2, 4) \).

Part 2: Evaluate \( f(1) \) and \( f(4) \)

  1. Finding \( f(1) \):

    • Since \( 1 \leq 2 \), use \( f(x) = 4x - 4 \): \[ f(1) = 4(1) - 4 = 4 - 4 = 0 \]

  2. Finding \( f(4) \):

    • Since \( 4 > 2 \), use \( f(x) = x^2 \): \[ f(4) = 4^2 = 16 \]

So:

  • \( f(1) = 0 \)
  • \( f(4) = 16 \)

Part 3: Domain of the Function

The function combines two pieces:

  • For \( x \leq 2 \), \( f(x) \) is defined.
  • For \( x > 2 \), \( f(x) \) is also defined.

Thus, the domain of the function is all real numbers since both pieces cover all possible values of \( x \).

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

Summary of Answers

  • Part 1: The correct graph is D.
  • Part 2: \( f(1) = 0 \) and \( f(4) = 16 \).
  • Part 3: The domain of the function is D. all real numbers.