Question

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