Find f left parenthesis 2 right parenthesis and f left parenthesis 5 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 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).
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 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).
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 2 right parenthesis and f left parenthesis 5 right parenthesis

Answer 1

To solve this problem, we first need to analyze the piecewise function given, which is defined as follows:

\[ f(x) = \begin{cases} 7x - 12 & \text{if } x \leq 3 \ x^2 & \text{if } x > 3 \end{cases} \]

Part a: Finding the graphs

Now, let's evaluate the possible graphs based on the description given.

1. For \( x \leq 3 \), the function \( f(x) = 7x - 12 \) is a linear equation, which means it will graph as a straight line. To find the y-intercept when \( x = 3 \): \[ f(3) = 7(3) - 12 = 21 - 12 = 9. \] So the point (3, 9) should belong to this portion.

2. For \( x > 3 \), the function \( f(x) = x^2 \) represents a quadratic function, which is a parabola. The value at \( x = 4 \) would be: \[ f(4) = 4^2 = 16. \]

Using this information, we can determine which graph is correct:

• The point (3, 9) is a closed endpoint since it is included in the first piece, and the curve should start from (3, 9) and continue as a parabola rising to the right typical for \( x^2 \).

Examining the Given Options:

• Option A: (3, 9) is included; the ray passes through (2, 2) and the curve is (3, 9) to the right. This fits.
• Option B: The point (3, 5) does not match our function. Therefore, it is incorrect.
• Option C: (3, 9) is open, which is incorrect because the point is closed in our function.
• Option D: No information provided.

The correct graph is A.

Part b: Finding \( f(2) \) and \( f(5) \)

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

   • Since \( 2 \leq 3 \), we use the first piece: \[ f(2) = 7(2) - 12 = 14 - 12 = 2. \]

2. Finding \( f(5) \):

   • Since \( 5 > 3 \), we use the second piece: \[ f(5) = 5^2 = 25. \]

Summary of results:

• \( f(2) = 2 \)
• \( f(5) = 25 \)

Part c: Domain of the function

The domain of the given piecewise function \( f(x) \) includes all real numbers since there are no restrictions stated on \( x \). Thus:

• Domain: \( (-\infty, \infty) \)

This concludes the questions posed about the function.