Question

Which equation is a function of x?
x = 5
x = y squared + 9
x squared = y
x squared = y squared + 16

Answer 1

To determine which equations represent functions of \( x \), we can analyze each equation to see if every \( x \) value corresponds to exactly one \( y \) value.

1. **\( x = 5 \)**: This does not define \( y \) in terms of \( x \), but instead states that \( x \) is always 5. It does not define a relationship between \( x \) and \( y \) in the traditional sense (function as \( y \) in terms of \( x \)).

2. **\( x = y^2 + 9 \)**: We can rearrange this to solve for \( y \):
\[
y^2 = x - 9
\]
This gives us:
\[
y = \pm \sqrt{x - 9}
\]
Since this equation yields two possible \( y \) values (one positive and one negative for each \( x \) greater than or equal to 9), it does not define \( y \) as a function of \( x \).

3. **\( x^2 = y \)**: We can solve this for \( y \):
\[
y = x^2
\]
Here, every value of \( x \) corresponds to one unique \( y \) (since \( y \) is defined as the square of \( x \)). Thus, this equation does define a function of \( x \).

4. **\( x^2 = y^2 + 16 \)**: Rearranging gives us:
\[
y^2 = x^2 - 16
\]
This means:
\[
y = \pm \sqrt{x^2 - 16}
\]
Similar to the second case, for \( x \) values greater than or equal to 4 (or \( x \leq -4 \)), there are two possible \( y \) values for each \( x \), so this does not define \( y \) as a function of \( x \).

**Conclusion**: The only equation that defines \( y \) as a function of \( x \) is \( x^2 = y \) (or \( y = x^2 \)).