Question
Use the table to describe the intervals over which f(x)equals=15xsquared2 is increasing and decreasing.
x
f(x)equals=15xsquared2
(x,y)
minus−2
60
(minus−2,60)
minus−1
15
(minus−1,15)
0
0
(0,0)
1
15
(1,15)
2
60
(2,60)
Question content area bottom
Part 1
The function f(x) is increasing over the interval enter your response here.
(Simplify your answer. Type an inequality.)
x
f(x)equals=15xsquared2
(x,y)
minus−2
60
(minus−2,60)
minus−1
15
(minus−1,15)
0
0
(0,0)
1
15
(1,15)
2
60
(2,60)
Question content area bottom
Part 1
The function f(x) is increasing over the interval enter your response here.
(Simplify your answer. Type an inequality.)
Answers
Answer
Use the table to describe the intervals over which f(x)equals=15xsquared2 is increasing and decreasing.
x
f(x)=15x^2
(x,y)
−2
60
(−2,60)
−1
15
(−1,15)
0
0
(0,0)
1
15
(1,15)
2
60
(2,60)
Question content area bottom
Part 1
The function f(x) is increasing over the interval enter your response here.
(Simplify your answer. Type an inequality.)
x
f(x)=15x^2
(x,y)
−2
60
(−2,60)
−1
15
(−1,15)
0
0
(0,0)
1
15
(1,15)
2
60
(2,60)
Question content area bottom
Part 1
The function f(x) is increasing over the interval enter your response here.
(Simplify your answer. Type an inequality.)
Answered by
GPT-4o mini
To determine the intervals over which the function \( f(x) = 15x^2 \) is increasing or decreasing, we can analyze the given values in the table:
- \( f(-2) = 60 \)
- \( f(-1) = 15 \)
- \( f(0) = 0 \)
- \( f(1) = 15 \)
- \( f(2) = 60 \)
From these values, we can observe the following behaviors:
1. As \( x \) increases from \(-2\) to \(-1\):
- \( f(-2) = 60 \) which is greater than \( f(-1) = 15 \).
- Therefore, the function is **decreasing** on the interval \((-2, -1)\).
2. As \( x \) increases from \(-1\) to \(0\):
- \( f(-1) = 15 \) which is greater than \( f(0) = 0 \).
- Therefore, the function is still **decreasing** on the interval \((-1, 0)\).
3. As \( x \) increases from \(0\) to \(1\):
- \( f(0) = 0 \) which is less than \( f(1) = 15 \).
- Therefore, the function is **increasing** on the interval \((0, 1)\).
4. As \( x \) increases from \(1\) to \(2\):
- \( f(1) = 15 \) which is less than \( f(2) = 60 \).
- Therefore, the function is also **increasing** on the interval \((1, 2)\).
In summary, the function is increasing over the intervals \((0, 1)\) and \((1, 2)\).
Combining these intervals, we can say that:
- The function \( f(x) \) is increasing over the interval \( (0, 2) \).
So, your answer for the interval where the function is increasing is:
\[
\text{The function } f(x) \text{ is increasing over the interval } (0, 2).
\]
- \( f(-2) = 60 \)
- \( f(-1) = 15 \)
- \( f(0) = 0 \)
- \( f(1) = 15 \)
- \( f(2) = 60 \)
From these values, we can observe the following behaviors:
1. As \( x \) increases from \(-2\) to \(-1\):
- \( f(-2) = 60 \) which is greater than \( f(-1) = 15 \).
- Therefore, the function is **decreasing** on the interval \((-2, -1)\).
2. As \( x \) increases from \(-1\) to \(0\):
- \( f(-1) = 15 \) which is greater than \( f(0) = 0 \).
- Therefore, the function is still **decreasing** on the interval \((-1, 0)\).
3. As \( x \) increases from \(0\) to \(1\):
- \( f(0) = 0 \) which is less than \( f(1) = 15 \).
- Therefore, the function is **increasing** on the interval \((0, 1)\).
4. As \( x \) increases from \(1\) to \(2\):
- \( f(1) = 15 \) which is less than \( f(2) = 60 \).
- Therefore, the function is also **increasing** on the interval \((1, 2)\).
In summary, the function is increasing over the intervals \((0, 1)\) and \((1, 2)\).
Combining these intervals, we can say that:
- The function \( f(x) \) is increasing over the interval \( (0, 2) \).
So, your answer for the interval where the function is increasing is:
\[
\text{The function } f(x) \text{ is increasing over the interval } (0, 2).
\]