Question

find $(f circ g)(10)$ in simplified form.
$f(x)$
$h(x) = 2x + 5$

Explanation:

Step1: Find $g(10)$ from table

From the table, when $x=10$, $g(10)=4$.

Step2: Find $f(g(10))=f(4)$

From the graph of $f(x)$, the parabola intersects the x-axis at $x=3$ and $x=7$, so its vertex is at $x=\frac{3+7}{2}=5$. Let the quadratic be $f(x)=a(x-3)(x-7)$. Using the y-intercept (when $x=0$, $f(0)=10$):
$10=a(0-3)(0-7) \implies 10=21a \implies a=\frac{10}{21}$
Substitute $x=4$:
$f(4)=\frac{10}{21}(4-3)(4-7)=\frac{10}{21}(1)(-3)=-\frac{30}{21}=-\frac{10}{7}$
Or directly read from the graph: at $x=4$, $f(4)=-2$ (matches the vertex offset, as vertex at $x=5$, $f(5)=-3$, so $f(4)=-2$)

Answer:

$-2$