Question

cms 2025 - 2026 nc math 4 initial credit composition of functions and modeling analyzing function composition the compositions $f(g(x))$ and $g(f(x))$ of functions $f$ and $g$ are shown on the graph. which statements describe the compositions? choose four correct answers. $f(g(0)) = 5$ and $g(f(-2.5)) = 5$. $f(g(x)) = g(f(x))$ for at least one value of $x$. the composition of $f$ and $g$ is commutative. both $f(g(x))$ and $g(f(x))$ have the same

Explanation:

Step1: Verify first statement

From the graph:

• For $f(g(0))$: $x=0$ on $f(g(x))$ gives $y=5$.
• For $g(f(-2.5))$: $x=-2.5$ on $g(f(x))$ gives $y=5$.

So $f(g(0))=5$ and $g(f(-2.5))=5$ is true.

Step2: Verify second statement

The two graphs intersect at at least one point, meaning there exists an $x$ where $f(g(x))=g(f(x))$. This statement is true.

Step3: Verify third statement

Functions are commutative if $f(g(x))=g(f(x))$ for all $x$. The graphs are not identical, so composition is not commutative. This statement is false.

Step4: Verify fourth statement

From the graph, both $f(g(x))$ and $g(f(x))$ have the same maximum $y$-value of 5, so they share the same maximum value. This statement is true.

(Note: Assuming the fourth statement is "Both $f(g(x))$ and $g(f(x))$ have the same maximum value" as implied by the graph, since the full text is cut off. Combined with the first two statements, this gives four correct answers, as required.)

Answer:

1. $f(g(0)) = 5$ and $g(f(-2.5)) = 5$
2. $f(g(x)) = g(f(x))$ for at least one value of $x$
3. Both $f(g(x))$ and $g(f(x))$ have the same maximum value

(Note: A fourth correct statement would be inferred from the full problem, but based on visible content, these three plus the implied fourth about shared maximum are the valid selections.)