Answer :
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 point means in the context of a function.
A point [tex]\((x, y)\)[/tex] being on the graph of a function [tex]\(f(x)\)[/tex] indicates that when the input [tex]\(x\)[/tex] is substituted into the function [tex]\(f\)[/tex], the output is [tex]\(y\)[/tex]. In other words, if the point [tex]\((4, 5)\)[/tex] is on the graph, it means:
[tex]\[ f(4) = 5 \][/tex]
So, if we have the coordinates [tex]\((4, 5)\)[/tex], we substitute [tex]\(x = 4\)[/tex] into the function and get [tex]\(f(4)\)[/tex], which should equal 5.
Now, let's examine the given options:
1. [tex]\( f(5) = 4 \)[/tex]
- This states that when the input of the function is 5, the output is 4. There is no information given that supports this statement, so this is not necessarily true.
2. [tex]\( f(5, 4) = 9 \)[/tex]
- This suggests a function of two variables, which doesn't align with the provided point indication ([tex]\(4, 5\)[/tex]) on a single-variable function graph. So this is not relevant here.
3. [tex]\( f(4) = 5 \)[/tex]
- This directly matches our analysis. It states that when the input is 4, the output is 5, which is exactly what the point [tex]\((4, 5)\)[/tex] signifies.
4. [tex]\( f(5, 4) = 1 \)[/tex]
- Similar to option 2, this is applicable to a function of two variables, which is not the case here.
Given the point [tex]\((4, 5)\)[/tex] on the graph, the correct equation that must be true is:
[tex]\[ f(4) = 5 \][/tex]
Thus, the correct choice is:
[tex]\[ f(4) = 5 \][/tex]
A point [tex]\((x, y)\)[/tex] being on the graph of a function [tex]\(f(x)\)[/tex] indicates that when the input [tex]\(x\)[/tex] is substituted into the function [tex]\(f\)[/tex], the output is [tex]\(y\)[/tex]. In other words, if the point [tex]\((4, 5)\)[/tex] is on the graph, it means:
[tex]\[ f(4) = 5 \][/tex]
So, if we have the coordinates [tex]\((4, 5)\)[/tex], we substitute [tex]\(x = 4\)[/tex] into the function and get [tex]\(f(4)\)[/tex], which should equal 5.
Now, let's examine the given options:
1. [tex]\( f(5) = 4 \)[/tex]
- This states that when the input of the function is 5, the output is 4. There is no information given that supports this statement, so this is not necessarily true.
2. [tex]\( f(5, 4) = 9 \)[/tex]
- This suggests a function of two variables, which doesn't align with the provided point indication ([tex]\(4, 5\)[/tex]) on a single-variable function graph. So this is not relevant here.
3. [tex]\( f(4) = 5 \)[/tex]
- This directly matches our analysis. It states that when the input is 4, the output is 5, which is exactly what the point [tex]\((4, 5)\)[/tex] signifies.
4. [tex]\( f(5, 4) = 1 \)[/tex]
- Similar to option 2, this is applicable to a function of two variables, which is not the case here.
Given the point [tex]\((4, 5)\)[/tex] on the graph, the correct equation that must be true is:
[tex]\[ f(4) = 5 \][/tex]
Thus, the correct choice is:
[tex]\[ f(4) = 5 \][/tex]
