Question

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 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**.