https://file.aiquickdraw.com/mathbot/file/17266827075990a1a9wn5.jpg

use this graph to answer all these questions, A)  What is the domain of k(x)k(x)? Give your answer using interval notation.    

B)  What is the range of k(x)k(x)? Give your answer using interval notation.    

C) What is/are the xx-intercepts of k(x)k(x)? Give your answer(s) in coordinate form (i.e. as ordered pairs), separating multiple answers with commas as needed.  

D) The zeros of k(x)k(x)  are  

E) k(9)k(9)  = 

F)  k(−7)k(-7)  = Correct

G)  Is k(5)k(5)  positive, negative, or 0?

positive

negative

0

Incorrect

H)  How many times does the line y=4y=4  intersect y=k(x)y=k(x) ? Correct

I)  For what value(s) of xx  does k(x)=5k(x)=5 ?  

J)  Give the interval(s) of xx such that k(x) > 0. Use the union symbol ∪∪  between multiple intervals.   

K) Give the interval(s) of xx such that k(x) < 0. Use the union symbol ∪∪ between multiple intervals.   

Let's analyze the graph to answer each of your questions step by step.

### A) **Domain of \( k(x) \)**

The domain refers to the set of all possible \( x \)-values for which the function \( k(x) \) is defined. Looking at the graph, \( k(x) \) starts from \( x = -7 \) and extends to \( x = 10 \).

**Domain:** \[ -7, 10 \]

In interval notation:
\[
\boxed{[-7, 10]}
\]

### B) **Range of \( k(x) \)**

The range is the set of all possible \( y \)-values (output values) for the function \( k(x) \). From the graph, the lowest point is around \( y = -2 \), and the highest point is around \( y = 6 \).

**Range:** \[ -2, 6 \]

In interval notation:
\[
\boxed{[-2, 6]}
\]

### C) **\( x \)-intercepts of \( k(x) \)**

The \( x \)-intercepts are the points where the function crosses the \( x \)-axis, i.e., where \( k(x) = 0 \). From the graph, these occur at \( x = -6 \), \( x = 0 \), and \( x = 8 \).

**\( x \)-intercepts (as ordered pairs):**
\[
\boxed{(-6, 0), (0, 0), (8, 0)}
\]

### D) **Zeros of \( k(x) \)**

The zeros of \( k(x) \) are the same as the \( x \)-intercepts, where \( k(x) = 0 \). Thus, the zeros are \( x = -6 \), \( x = 0 \), and \( x = 8 \).

**Zeros:**
\[
\boxed{-6, 0, 8}
\]

### E) **\( k(9) \)**

From the graph, when \( x = 9 \), \( k(9) \) corresponds to the \( y \)-value at that point. The graph shows that \( k(9) \approx 2 \).

\[
\boxed{k(9) = 2}
\]

### F) **\( k(-7) \)**

From the graph, when \( x = -7 \), \( k(-7) = 0 \), as the graph passes through the \( x \)-axis at this point.

\[
\boxed{k(-7) = 0}
\]

### G) **Is \( k(5) \) positive, negative, or 0?**

At \( x = 5 \), the graph is below the \( x \)-axis, meaning \( k(5) \) is negative.

\[
\boxed{\text{negative}}
\]

### H) **How many times does the line \( y = 4 \) intersect \( y = k(x) \)?**

The line \( y = 4 \) intersects the graph of \( k(x) \) in two places.

\[
\boxed{2}
\]

### I) **For what value(s) of \( x \) does \( k(x) = 5 \)?**

From the graph, \( k(x) = 5 \) occurs at approximately \( x = -5 \) and \( x = 6 \).

\[
\boxed{x = -5, 6}
\]

### J) **Interval(s) where \( k(x) > 0 \)**

From the graph, \( k(x) > 0 \) occurs between:
- \( x = -7 \) and \( x = -6 \)
- \( x = 0 \) and \( x = 8 \)
- \( x = 9 \) and \( x = 10 \)

In interval notation:
\[
\boxed{(-7, -6) \cup (0, 8) \cup (9, 10)}
\]

### K) **Interval(s) where \( k(x) < 0 \)**

From the graph, \( k(x) < 0 \) occurs between:
- \( x = -6 \) and \( x = 0 \)
- \( x = 8 \) and \( x = 9 \)

In interval notation:
\[
\boxed{(-6, 0) \cup (8, 9)}
\]

---

Would you like any further details on these calculations? 

Here are 5 related questions you might find useful:

1. How would the graph change if \( k(x) \) were shifted upwards by 2 units?
2. What are the critical points of \( k(x) \), and what do they represent?
3. Can you find the intervals where \( k(x) \) is increasing or decreasing?
4. How can you calculate the slope of the tangent line at specific points on this graph?
5. What are the local maxima and minima of \( k(x) \)?

**Tip:** Understanding how to read a graph’s domain and range is crucial when analyzing real-world applications, like economics or physics, where functions often describe systems with limits.

A) What is the domain of k(x)? Give your answer using interval notation.
B) What is the range of k(x)? Give your answer using interval notation.
C) What is/are the x-intercepts of k(x)? Give your answer(s) in coordinate form.
D) The zeros of k(x) are?
E) k(9) = ?
F) k(-7) = ?
G) Is k(5) positive, negative, or 0?
H) How many times does the line y=4 intersect y=k(x)?
I) For what value(s) of x does k(x)=5?
J) Give the interval(s) where k(x) > 0.
K) Give the interval(s) where k(x) < 0.

Analyze the graph of k(x) to determine its domain, range, zeros, and x-intercepts. This detailed breakdown also covers points like k(9), k(-7), and intervals where k(x) is greater or less than 0. Suitable for high school students studying algebra and functions.

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