Composing a table function with a formula

Work from the inside out. The notation $f(g(3))$ means: find $g(3)$, then use that number as the input to $f$.

Inner step, from the table: the row with $x=3$ gives $g(3)=-2$.

Outer step, from the formula $f(x)=3^x+x^2$: $$f(-2)=3^{-2}+(-2{)}^2=\frac{1}{9}+4=\frac{1}{9}+\frac{36}{9}=\frac{37}{9}$$

So $f(g(3))=\frac{37}{9}$, choice B.

A negative exponent gives a reciprocal, so $3^{-2}$ is $\frac{1}{9}$ and not $-9$, and squaring $-2$ gives positive $4$.

Each choice

• A. Multiplying instead of composing gives this: $f(3)=27+9=36$ and $36\cdot g(3)=36(-2)=-72$. Composition feeds one output into the next function; it does not multiply them.
• B (correct). The table gives $g(3)=-2$, and $f(-2)=3^{-2}+4=\frac{37}{9}$.
• C. This runs the composition backwards: $f(3)=36$, and the table then gives $g(36)=9$. That is $g(f(3))$, which is a different function from $f(g(3))$.
• D. Looking up 3 in the output column finds $g(4)=3$, and then $f(4)=81+16=97$. The 3 in $g(3)$ is an input, so it belongs in the $x$ column.