To determine which equation must be true given that the point [tex]\((4, 5)\)[/tex] is on the graph of a function, let's break down the concepts involved.
When a point [tex]\((a, b)\)[/tex] is on the graph of a function [tex]\(f(x)\)[/tex], it means that the function [tex]\(f\)[/tex] evaluated at [tex]\(x = a\)[/tex] equals [tex]\(b\)[/tex]. In other words, [tex]\(f(a) = b\)[/tex].
Given the point [tex]\((4, 5)\)[/tex]:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 5\)[/tex]
To determine the correct equation, we need to find the expression that represents [tex]\(f(4) = 5\)[/tex].
Now let's analyze each equation provided:
1. [tex]\( f(5) = 4 \)[/tex]:
- This implies that if we evaluate the function [tex]\(f\)[/tex] at [tex]\(x = 5\)[/tex], we get the result 4.
- However, this does not correspond to the given point [tex]\((4, 5)\)[/tex].
2. [tex]\( f(5, 4) = 9 \)[/tex]:
- This implies that the function [tex]\(f\)[/tex] takes two inputs and outputs 9.
- This notation is not relevant to our case since the given point is of the form [tex]\( (x, y) \)[/tex] where [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are interpreted typically in one-variable functions.
3. [tex]\( f(4) = 5 \)[/tex]:
- This directly translates to the point [tex]\((4, 5)\)[/tex] on the function [tex]\(f\)[/tex].
- Evaluating the function at [tex]\( x = 4 \)[/tex] yields 5, exactly what we are given.
4. [tex]\( f(5, 4) = 1 \)[/tex]:
- Similar to the second option, this implies a two-variable function that outputs 1, which is not relevant to our case.
Clearly, the equation that must be true is:
[tex]\[ f(4) = 5 \][/tex]
Thus, the correct equation is:
[tex]\[ \boxed{f(4) = 5} \][/tex]
