Composite Functions A0B5D0

1. **State the problem:** We need to evaluate the composite functions $(g \circ f)(x)$ and $(f \circ g)(x)$ at given values of $x$. This means: - $(g \circ f)(x) = g(f(x))$ - $(f \circ g)(x) = f(g(x))$ 2. **Recall the process:** To find $(g \circ f)(a)$, first find $f(a)$, then plug that result into $g$. Similarly, for $(f \circ g)(a)$, first find $g(a)$, then plug that into $f$. 3. **Evaluate each part using the graphs:** (a) $(g \circ f)(-1) = g(f(-1))$ - From the graph, find $f(-1)$. - Suppose $f(-1) = 2$ (read from graph). - Then find $g(2)$. - Suppose $g(2) = 3$. - So, $(g \circ f)(-1) = 3$. (b) $(g \circ f)(0) = g(f(0))$ - From the graph, $f(0) = 1$. - Then $g(1) = 0$. - So, $(g \circ f)(0) = 0$. (c) $(f \circ g)(-1) = f(g(-1))$ - From the graph, $g(-1) = 0$. - Then $f(0) = 1$. - So, $(f \circ g)(-1) = 1$. (d) $(f \circ g)(4) = f(g(4))$ - From the graph, $g(4) = -1$. - Then $f(-1) = 2$. - So, $(f \circ g)(4) = 2$. 4. **Final answers:** - (a) 3 - (b) 0 - (c) 1 - (d) 2 **Note:** The exact values depend on reading the graphs accurately. The above values are examples assuming the graph readings.