Use the graph to find the following.
-6
6
12
6
12
18
24
-6
x
y



A coordinate system has a horizontal x-axis labeled from negative 6 to 12 in increments of 1 and a vertical y-axis labeled from negative 6 to 26 in increments of 1. A horizontal ray starts at point (0, 24) and extends to the left. A curve falls from left to right starting at the point (0, 24) and passing through the point (3, 3) to the point (5, negative 1). The curve then rises from left to right passing through the point (7, 3).
​(a) the domain of f
​(b) the range of f  
​(c) the​ x-intercepts
​(d) the​ y-intercept
​(e) intervals on which f is increasing
​(f) intervals on which f is decreasing
​(g) intervals on which f is constant
​(h) the number at which f has a relative minimum
​(i) the relative minimum of f   
​(j) ​f(0​)
​(k) The values of x for which ​f(x)equals3
​(l) Is f​ even, odd or​ neither?
Question content area bottom
Part 1
​(a) What is the domain of​ f?
  
left parenthesis negative infinity comma infinity right parenthesis ​(Type your answer in interval​ notation.)
Part 2
​(b) What is the range of​ f?
  
left bracket negative 1 comma infinity right parenthesis  ​(Type your answer in interval​ notation.)
Part 3
​(c) What are the​ x-intercepts of the​ function?
The left​ x-intercept of the function is 4 and the right​ x-intercept is 
  
6.
Part 4
​(d) What is the​ y-intercept?
The​ y-intercept of the function is 
  
24.
Part 5
​(e)  Over what interval is f​ increasing?
  
enter your response here  ​(Type your answer in interval​ notation.)


Let's go through the parts of the problem one by one based on the graph described.

### (a) Domain of \( f \)
The domain of a function is the set of all possible \( x \)-values for which the function is defined. The graph shows that the function has values starting from \( x = -\infty \) to \( x = \infty \), as it includes a ray extending indefinitely to the left. Hence, the domain is:

\[
(-\infty, \infty)
\]

### (b) Range of \( f \)
The range of a function is the set of all possible \( y \)-values. From the graph, the lowest \( y \)-value occurs at \( y = -1 \) (at \( x = 5 \)). The function increases upwards indefinitely after that. So, the range is:

\[
[-1, \infty)
\]

### (c) \( x \)-intercepts of \( f \)
The \( x \)-intercepts are the points where the function crosses the \( x \)-axis. Based on the graph, the curve crosses the \( x \)-axis at \( x = 4 \) and \( x = 6 \). Therefore, the \( x \)-intercepts are:

\[
x = 4 \quad \text{and} \quad x = 6
\]

### (d) \( y \)-intercept of \( f \)
The \( y \)-intercept is the point where the function crosses the \( y \)-axis. The graph shows that the function crosses the \( y \)-axis at \( (0, 24) \). Therefore, the \( y \)-intercept is:

\[
y = 24
\]

### (e) Intervals on which \( f \) is increasing
A function is increasing on intervals where its slope is positive. Based on the graph:

- The curve rises from \( x = 5 \) to \( x = 7 \).
- The ray extending to the left is horizontal, so it doesn't count as increasing.

Thus, \( f \) is increasing on the interval:

\[
(5, 7)
\]

Would you like more details or have any questions? Here are five related questions that extend the concepts:

1. What are the intervals where \( f \) is decreasing?
2. How do we find intervals where \( f \) is constant from a graph?
3. What is the significance of the relative minimum?
4. Can a function have multiple \( y \)-intercepts? Why or why not?
5. How does the behavior of the function indicate whether it is even, odd, or neither?

**Tip**: Always pay attention to the slope of a function to determine increasing or decreasing intervals.

Use the graph to find the following. A coordinate system has a horizontal x-axis labeled from negative 6 to 12 in increments of 1 and a vertical y-axis labeled from negative 6 to 26 in increments of 1. A horizontal ray starts at point (0, 24) and extends to the left. A curve falls from left to right starting at the point (0, 24) and passing through the point (3, 3) to the point (5, negative 1). The curve then rises from left to right passing through the point (7, 3). (a) the domain of f (b) the range of f (c) the x-intercepts (d) the y-intercept (e) intervals on which f is increasing (f) intervals on which f is decreasing (g) intervals on which f is constant (h) the number at which f has a relative minimum (i) the relative minimum of f (j) f(0​) (k) The values of x for which f(x) equals 3 (l) Is f​ even, odd, or neither?

Analyze the graph of a function to determine its domain, range, intercepts, intervals of increase and decrease, and symmetry. Understand how the function behaves from left to right, its minimum points, and how to interpret even, odd, or neither functions. Suitable for high school students (Grades 9-12).

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