Question
If h(x)
is the horizontal reflection of the function f(x)=x3
, which of the following tables can be the datasets for f(x)
and h(x)
?(1 point)
Responses
x f(x)
0 0
1 1
2 4
x h(x)
0 0
−1
1
−2
4
x f left parenthesis x right parenthesis 0 0 1 1 2 4 x h left parenthesis x right parenthesis 0 0 negative 1 1 negative 2 4
x f(x)
0 0
1 1
2 8
x h(x)
0 0
−1
−1
−2
−8
x f left parenthesis x right parenthesis 0 0 1 1 2 8 x h left parenthesis x right parenthesis 0 0 negative 1 negative 1 negative 2 negative 8
x f(x)
0 0
1 3
2 6
x h(x)
0 0
−1
3
−2
6
x f left parenthesis x right parenthesis 0 0 1 3 2 6 x h left parenthesis x right parenthesis 0 0 negative 1 3 negative 2 6
x f(x)
0 0
1 1
2 8
x h(x)
0 0
−1
1
−2
8
x f left parenthesis x right parenthesis 0 0 1 1 2 8 x h left parenthesis x right parenthesis 0 0 negative 1 1 negative 2 8
Remaining Attempts : 3
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is the horizontal reflection of the function f(x)=x3
, which of the following tables can be the datasets for f(x)
and h(x)
?(1 point)
Responses
x f(x)
0 0
1 1
2 4
x h(x)
0 0
−1
1
−2
4
x f left parenthesis x right parenthesis 0 0 1 1 2 4 x h left parenthesis x right parenthesis 0 0 negative 1 1 negative 2 4
x f(x)
0 0
1 1
2 8
x h(x)
0 0
−1
−1
−2
−8
x f left parenthesis x right parenthesis 0 0 1 1 2 8 x h left parenthesis x right parenthesis 0 0 negative 1 negative 1 negative 2 negative 8
x f(x)
0 0
1 3
2 6
x h(x)
0 0
−1
3
−2
6
x f left parenthesis x right parenthesis 0 0 1 3 2 6 x h left parenthesis x right parenthesis 0 0 negative 1 3 negative 2 6
x f(x)
0 0
1 1
2 8
x h(x)
0 0
−1
1
−2
8
x f left parenthesis x right parenthesis 0 0 1 1 2 8 x h left parenthesis x right parenthesis 0 0 negative 1 1 negative 2 8
Remaining Attempts : 3
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Answers
GPT-4o mini
To find the horizontal reflection of the function \( f(x) = x^3 \), we need to consider how the reflection affects the graph. A horizontal reflection across the y-axis can be represented by the function \( h(x) = f(-x) \).
Given \( f(x) = x^3 \), we can find \( h(x) \):
\[
h(x) = f(-x) = (-x)^3 = -x^3
\]
Now we need to find the output values of \( h(x) \) that correspond to the input values in \( f(x) \).
### Original Function \( f(x) = x^3 \)
- For \( x = 0 \): \( f(0) = 0^3 = 0 \)
- For \( x = 1 \): \( f(1) = 1^3 = 1 \)
- For \( x = 2 \): \( f(2) = 2^3 = 8 \)
Thus, we have:
- \( f(0) = 0 \)
- \( f(1) = 1 \)
- \( f(2) = 8 \)
### Reflected Function \( h(x) = -x^3 \)
- For \( x = 0 \): \( h(0) = -0^3 = 0 \)
- For \( x = -1 \): \( h(-1) = -(-1)^3 = 1 \)
- For \( x = -2 \): \( h(-2) = -(-2)^3 = 8 \)
Thus, we have:
- \( h(0) = 0 \)
- \( h(-1) = 1 \)
- \( h(-2) = 8 \)
### Evaluating Responses
Now we will evaluate the provided tables based on this information:
1. **First Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 4 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 1 \\
1 & -1 \\
-2 & 4 \\
\end{array}
\]
- \( f(2) = 4 \) (should be 8).
- This table is **incorrect**.
2. **Second Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & -1 \\
-2 & -8 \\
\end{array}
\]
- \( h(-1) \) should be 1 (not -1), and \( h(-2) \) should be 8 (not -8).
- This table is **incorrect**.
3. **Third Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 3 \\
2 & 6 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 3 \\
-2 & 6 \\
\end{array}
\]
- Values of \( f(1) \) and \( f(2) \) are incorrect. \( f(1) \) should be 1; likewise, \( f(2) \) should be 8.
- This table is **incorrect**.
4. **Fourth Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 1 \\
-2 & 8 \\
\end{array}
\]
- \( h(-1) \) should be 1 (correct) and \( h(-2) \) should be 8 (not correct, should be -8).
- This table is **incorrect**.
### Summary
None of the tables presented correctly matches the reflections of the functions \( f(x) = x^3 \) and its horizontal reflection \( h(x) = -x^3 \). All options contain inaccuracies or misrepresent either \( f(x) \) or \( h(x) \).
Given \( f(x) = x^3 \), we can find \( h(x) \):
\[
h(x) = f(-x) = (-x)^3 = -x^3
\]
Now we need to find the output values of \( h(x) \) that correspond to the input values in \( f(x) \).
### Original Function \( f(x) = x^3 \)
- For \( x = 0 \): \( f(0) = 0^3 = 0 \)
- For \( x = 1 \): \( f(1) = 1^3 = 1 \)
- For \( x = 2 \): \( f(2) = 2^3 = 8 \)
Thus, we have:
- \( f(0) = 0 \)
- \( f(1) = 1 \)
- \( f(2) = 8 \)
### Reflected Function \( h(x) = -x^3 \)
- For \( x = 0 \): \( h(0) = -0^3 = 0 \)
- For \( x = -1 \): \( h(-1) = -(-1)^3 = 1 \)
- For \( x = -2 \): \( h(-2) = -(-2)^3 = 8 \)
Thus, we have:
- \( h(0) = 0 \)
- \( h(-1) = 1 \)
- \( h(-2) = 8 \)
### Evaluating Responses
Now we will evaluate the provided tables based on this information:
1. **First Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 4 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 1 \\
1 & -1 \\
-2 & 4 \\
\end{array}
\]
- \( f(2) = 4 \) (should be 8).
- This table is **incorrect**.
2. **Second Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & -1 \\
-2 & -8 \\
\end{array}
\]
- \( h(-1) \) should be 1 (not -1), and \( h(-2) \) should be 8 (not -8).
- This table is **incorrect**.
3. **Third Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 3 \\
2 & 6 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 3 \\
-2 & 6 \\
\end{array}
\]
- Values of \( f(1) \) and \( f(2) \) are incorrect. \( f(1) \) should be 1; likewise, \( f(2) \) should be 8.
- This table is **incorrect**.
4. **Fourth Table:**
\[
\begin{array}{cc}
x & f(x) \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
\end{array}
\quad
\begin{array}{cc}
x & h(x) \\
0 & 0 \\
-1 & 1 \\
-2 & 8 \\
\end{array}
\]
- \( h(-1) \) should be 1 (correct) and \( h(-2) \) should be 8 (not correct, should be -8).
- This table is **incorrect**.
### Summary
None of the tables presented correctly matches the reflections of the functions \( f(x) = x^3 \) and its horizontal reflection \( h(x) = -x^3 \). All options contain inaccuracies or misrepresent either \( f(x) \) or \( h(x) \).