Answer :
To solve the problem of which equation must be true given that the point \((4, 5)\) is on the graph of the function, let's look at what this point represents on a graph of a function.
A point \((x, y)\) on the graph of a function \(f\) indicates that when \(x\) is input into the function \(f\), the output \(y\) is the result. This means \(f(x) = y\).
Given the point \((4, 5)\), the x-coordinate is \(4\) and the y-coordinate is \(5\). Thus, when \(4\) is the input \(x\) value for the function \(f\), the output \(y\) value should be \(5\).
This can be written as:
[tex]\[ f(4) = 5 \][/tex]
Looking through the given options:
1. \(f(5) = 4\): This option states that when the input is \(5\), the output is \(4\). This does not match our point \((4, 5)\) because it reverses the roles of the input and output.
2. \(f(5, 4) = 9\): This option is incorrectly formulated. It considers \(f\) as a function of two variables, which is not applicable in this context.
3. \(f(4) = 5\): This correctly states that when the input is \(4\), the output is \(5\), which is consistent with the point \((4, 5)\).
4. \(f(5, 4) = 1\): Similar to option 2, this incorrectly considers \(f\) as a function of two variables.
The correct equation that must be true given the point \((4, 5)\) is:
[tex]\[ f(4) = 5 \][/tex]
Thus, the correct answer is:
[tex]\( \boxed{f(4) = 5} \)[/tex]
A point \((x, y)\) on the graph of a function \(f\) indicates that when \(x\) is input into the function \(f\), the output \(y\) is the result. This means \(f(x) = y\).
Given the point \((4, 5)\), the x-coordinate is \(4\) and the y-coordinate is \(5\). Thus, when \(4\) is the input \(x\) value for the function \(f\), the output \(y\) value should be \(5\).
This can be written as:
[tex]\[ f(4) = 5 \][/tex]
Looking through the given options:
1. \(f(5) = 4\): This option states that when the input is \(5\), the output is \(4\). This does not match our point \((4, 5)\) because it reverses the roles of the input and output.
2. \(f(5, 4) = 9\): This option is incorrectly formulated. It considers \(f\) as a function of two variables, which is not applicable in this context.
3. \(f(4) = 5\): This correctly states that when the input is \(4\), the output is \(5\), which is consistent with the point \((4, 5)\).
4. \(f(5, 4) = 1\): Similar to option 2, this incorrectly considers \(f\) as a function of two variables.
The correct equation that must be true given the point \((4, 5)\) is:
[tex]\[ f(4) = 5 \][/tex]
Thus, the correct answer is:
[tex]\( \boxed{f(4) = 5} \)[/tex]
