Question
Which relation is a function of x?
{(1, 2), (7, 6), (3, 2), (1, 0), (5, 6)}
A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 0, 0, 0. Column 2 is labeled y with entries 2, negative 6, 9, negative 7.
x = 3 y squared minus 7
On a coordinate plane, a graph curves up, then curves down, and then curves up again.
{(1, 2), (7, 6), (3, 2), (1, 0), (5, 6)}
A 2-column table with 4 rows. Column 1 is labeled x with entries 0, 0, 0, 0. Column 2 is labeled y with entries 2, negative 6, 9, negative 7.
x = 3 y squared minus 7
On a coordinate plane, a graph curves up, then curves down, and then curves up again.
Answers
GPT-4o mini
To determine which of the given relations is a function of \( x \), we need to analyze each option to see if every input \( x \) corresponds to exactly one output \( y \).
1. **Set of Ordered Pairs:**
\[
\{(1, 2), (7, 6), (3, 2), (1, 0), (5, 6)\}
\]
In this set, the input \( x = 1 \) corresponds to two different outputs: \( y = 2 \) and \( y = 0 \). Since there are multiple outputs for the same input, this relation is **not a function**.
2. **2-column Table:**
```
x | y
-----
0 | 2
0 | -6
0 | 9
0 | -7
```
Here, all entries in the first column (all \( x \) values) are the same (0), but they correspond to different \( y \) values (2, -6, 9, -7). Hence, this relation is **not a function**.
3. **Equation:**
\[
x = 3y^2 - 7
\]
To check if this is a function, we can rewrite it in terms of \( y \):
\[
3y^2 = x + 7 \quad \Rightarrow \quad y^2 = \frac{x + 7}{3}
\]
This implies:
\[
y = \pm \sqrt{\frac{x + 7}{3}}
\]
Since each \( x \) value can give two possible \( y \) values (one positive and one negative), this is **not a function**.
4. **Graph Description:**
The description "a graph curves up, then curves down, and then curves up again" suggests that the graph has one or more turning points. A function cannot have multiple outputs (y-values) for the same input (x-value). Such a graph typically does not satisfy the vertical line test, meaning that it is **not a function**.
From all the options given, **none of the relations or descriptions provided are functions of \( x \)**, as they all fail to meet the primary criterion that each input must correspond to exactly one output.
1. **Set of Ordered Pairs:**
\[
\{(1, 2), (7, 6), (3, 2), (1, 0), (5, 6)\}
\]
In this set, the input \( x = 1 \) corresponds to two different outputs: \( y = 2 \) and \( y = 0 \). Since there are multiple outputs for the same input, this relation is **not a function**.
2. **2-column Table:**
```
x | y
-----
0 | 2
0 | -6
0 | 9
0 | -7
```
Here, all entries in the first column (all \( x \) values) are the same (0), but they correspond to different \( y \) values (2, -6, 9, -7). Hence, this relation is **not a function**.
3. **Equation:**
\[
x = 3y^2 - 7
\]
To check if this is a function, we can rewrite it in terms of \( y \):
\[
3y^2 = x + 7 \quad \Rightarrow \quad y^2 = \frac{x + 7}{3}
\]
This implies:
\[
y = \pm \sqrt{\frac{x + 7}{3}}
\]
Since each \( x \) value can give two possible \( y \) values (one positive and one negative), this is **not a function**.
4. **Graph Description:**
The description "a graph curves up, then curves down, and then curves up again" suggests that the graph has one or more turning points. A function cannot have multiple outputs (y-values) for the same input (x-value). Such a graph typically does not satisfy the vertical line test, meaning that it is **not a function**.
From all the options given, **none of the relations or descriptions provided are functions of \( x \)**, as they all fail to meet the primary criterion that each input must correspond to exactly one output.