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.
0
6
-15
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 15 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, 2). A curve starts at the open point and rises from left to right at an increasing rate, passing through the point (4, 16).
B.
0
6
-15
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 15 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 2). 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
-15
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 15 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, 2). 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
-15
40
x
y

a. Graph f left parenthesis x right parenthesis equals
7 x minus 12
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 4 right parenthesis.
c. State the domain of the function.

Answer 1

To tackle this problem, we need to analyze the piecewise function given and then evaluate it at the specified points.

a. Choose the correct graph.

The piecewise function is defined as follows:

• \( f(x) = 7x - 12 \) if \( x \leq 3 \)
• \( f(x) = x^2 \) if \( x > 3 \)

Based on this definition:

1. For \( x \leq 3 \), the function is linear (specifically, a line with a slope of 7).
2. For \( x > 3 \), the function is a quadratic function that will be a parabola opening upwards.

From the graph descriptions, we can determine if they match the given piecewise function:

• Graph C describes a ray starting at the point (3, 9) and falling to the left, while a curve starts from (3, 9) and rises. This aligns with the description, as \( f(3) = 7(3)-12 = 9 \) and the next part for \( x > 3 \) gives \( f(4) = 4^2 = 16 \).

Thus, Graph C is correct.

b. Find \( f(1) \) and \( f(4) \).

1. Finding \( f(1) \): Since \( 1 \leq 3 \), we use the first part of the piecewise function: \[ f(1) = 7(1) - 12 = 7 - 12 = -5 \]

2. Finding \( f(4) \): Since \( 4 > 3 \), we use the second part of the piecewise function: \[ f(4) = 4^2 = 16 \]

c. State the domain of the function.

The function is defined for all real numbers since there are no restrictions on the input values for either piece of the function. Hence, the domain is:

\[ \text{Domain} = (-\infty, \infty) \]

Summary:

• Correct graph: Graph C
• \( f(1) = -5 \)
• \( f(4) = 16 \)
• Domain: \( (-\infty, \infty) \)