To determine which equation must be true given that the point [tex]\((4, 5)\)[/tex] is on the graph of the function, let’s analyze what this information implies:
1. A point [tex]\((a, b)\)[/tex] on the graph of a function [tex]\(f\)[/tex] indicates that when the input (or [tex]\(x\)[/tex]-value) is [tex]\(a\)[/tex], the output (or [tex]\(y\)[/tex]-value) is [tex]\(b\)[/tex]. In other words, [tex]\(f(a) = b\)[/tex].
In this specific case, the point [tex]\((4, 5)\)[/tex] means that when [tex]\(x = 4\)[/tex], [tex]\(f(x) = 5\)[/tex]. Therefore, the relationship between [tex]\(x\)[/tex] and [tex]\(f(x)\)[/tex] is expressed by the equation:
[tex]\[ f(4) = 5 \][/tex]
Let's verify this by checking the other given equations:
- [tex]\(f(5) = 4\)[/tex]: This would mean that when the input [tex]\(x\)[/tex] is 5, the output [tex]\(f(x)\)[/tex] is 4, which does not match [tex]\((4, 5)\)[/tex].
- [tex]\(f(5, 4) = 9\)[/tex]: This appears to be an equation involving two inputs, which is inconsistent with the given single variable function implied by the point [tex]\((4, 5)\)[/tex].
- [tex]\(f(5, 4) = 1\)[/tex]: Similar to the previous option, this involves a two-variable function, which is inconsistent.
Hence, the true and correct equation that must be true given the point [tex]\((4, 5)\)[/tex] is:
[tex]\[ f(4) = 5 \][/tex]
