Question: Evaluate each expression using the values given in the table. x − 3 − 2 − 1 0 1 2 3 f(x) − 5 − 4 − 3 − 1 3 4 5 g(x) 7 3 0 − 1 0 3 7 a. (f o g)(1) b. (f o g)(− 1) c. (g o f)(− 1) d. (g o f)(0) e. (g o g)(− 2) f. (f o f)(− 1)

1. **State the problem:** We need to evaluate composite functions such as $(f \circ g)(1)$, $(f \circ g)(-1)$, $(g \circ f)(-1)$, $(g \circ f)(0)$, $(g \circ g)(-2)$, and $(f \circ f)(-1)$ using the given tables for $f(x)$ and $g(x)$. 2. **Recall the definition of composite functions:** - $(f \circ g)(x) = f(g(x))$ means we first find $g(x)$, then plug that result into $f$. - Similarly, $(g \circ f)(x) = g(f(x))$. 3. **Evaluate each part step-by-step:** **a. $(f \circ g)(1) = f(g(1))$** - From the table, $g(1) = 0$. - Then $f(0) = -1$. - So, $(f \circ g)(1) = -1$. **b. $(f \circ g)(-1) = f(g(-1))$** - From the table, $g(-1) = 0$. - Then $f(0) = -1$. - So, $(f \circ g)(-1) = -1$. **c. $(g \circ f)(-1) = g(f(-1))$** - From the table, $f(-1) = -3$. - Then $g(-3) = 7$. - So, $(g \circ f)(-1) = 7$. **d. $(g \circ f)(0) = g(f(0))$** - From the table, $f(0) = -1$. - Then $g(-1) = 0$. - So, $(g \circ f)(0) = 0$. **e. $(g \circ g)(-2) = g(g(-2))$** - From the table, $g(-2) = 3$. - Then $g(3) = 7$. - So, $(g \circ g)(-2) = 7$. **f. $(f \circ f)(-1) = f(f(-1))$** - From the table, $f(-1) = -3$. - Then $f(-3) = -5$. - So, $(f \circ f)(-1) = -5$. **Final answers:** - a. $-1$ - b. $-1$ - c. $7$ - d. $0$ - e. $7$ - f. $-5$