Find f left parenthesis 1 right parenthesis and f left parenthesis 4 right parenthesis.

Question

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 open point (2, 4) and falls from right to left passing through the point (1, 1). A curve starts at the open point 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 3). 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 closed point (2, 0) and falls from right to left passing through the point (1, negative 3). 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).
D.
-1
5
-10
30
x
y

a. Graph f left parenthesis x right parenthesis equals
3 x minus 2
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.

Answer 1

To analyze the function provided:

The piecewise function is defined as follows:

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

Part (b): Find \( f(1) \) and \( f(4) \)

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

   • Since \( 1 \leq 2 \), we use the first part of the piecewise function: \[ f(1) = 3(1) - 2 = 3 - 2 = 1 \]

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

   • Since \( 4 > 2 \), we use the second part of the piecewise function: \[ f(4) = 4^2 = 16 \]

Thus, the results are:

• \( f(1) = 1 \)
• \( f(4) = 16 \)

Part (c): State the domain of the function

The domain of the function is the set of all \( x \) values for which the function is defined. Both parts of the piecewise function are defined for all real numbers. Therefore, the domain is: \[ \text{Domain: } (-\infty, \infty) \]

Part (a): Choose the correct graph

From the description of the graph:

1. The function \( f(1) = 1 \) and the graph needs to show the point (1, 1) and the ray on the left segment should be descending, consistent with the equation \( f(x) = 3x - 2 \).
2. For \( f(4) = 16 \), we need the point (4, 16) to be on the curve segment where \( f(x) = x^2 \).

Looking at the provided graphs:

• In option A, the ray starts at an open point (2, 4) and goes down to (1, 1), which matches our calculations.
• The other options seem to differ in how the segments are defined and the points (e.g., some start at (2, 0) or are positioned incorrectly).

Therefore, the correct answer is A.

Summary

• \( f(1) = 1 \)
• \( f(4) = 16 \)
• Domain: \( (-\infty, \infty) \)
• Correct graph option: A.