To convert the equation [tex]\( y - x^2 = 5 \)[/tex] into function notation, we need to follow these steps:
1. Isolate the variable [tex]\( y \)[/tex]: Start with the given equation:
[tex]\[
y - x^2 = 5
\][/tex]
Add [tex]\( x^2 \)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = x^2 + 5
\][/tex]
2. Replace [tex]\( y \)[/tex] with [tex]\( f(x) \)[/tex]: To express this in function notation, replace [tex]\( y \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x^2 + 5
\][/tex]
So, the correct way to write [tex]\( y - x^2 = 5 \)[/tex] in function notation is:
[tex]\[
f(x) = x^2 + 5
\][/tex]
Among the given options:
- [tex]\( \text{A } y = x^2 + 5 \)[/tex] shows [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], but it is not in function notation.
- [tex]\( \text{B } y = f(x)^2 + 5 \)[/tex] is incorrect because it suggests [tex]\( y \)[/tex] is equal to the square of a function plus 5.
- [tex]\( \text{C } f(x) = x^2 + 5 \)[/tex] correctly shows the function notation.
- [tex]\( \text{D } f(x) = y - 5 \)[/tex] is incorrect as it does not represent the original equation.
Therefore, the correct answer is:
[tex]\[
\text{C } f(x) = x^2 + 5
\][/tex]
