Question

evaluate each expression using the graphs of \\(y = f(x)\\) and \\(y = g(x)\\) shown.

(a) \\((g \circ f)(-1)\\)
(b) \\((g \circ f)(0)\\)
(c) \\((f \circ g)(-1)\\)
(d) \\((f \circ g)(4)\\)

Explanation:

Evaluate (g o f)(-1)

Using the Composite Functions and Graphical Function Evaluation knowledge points

$$ (g \circ f)(-1) = g(f(-1)) $$

From the graph of \(y = f(x)\) (the lower V-shaped curve):

$$ f(-1) = 1 $$

Now evaluate \(g(1)\) from the graph of \(y = g(x)\) (the upper curved graph):

$$ g(1) = 3 $$

Thus:

$$ (g \circ f)(-1) = 3 $$

Evaluate (g o f)(0)

Using the Composite Functions and Graphical Function Evaluation knowledge points

$$ (g \circ f)(0) = g(f(0)) $$

From the graph of \(y = f(x)\):

$$ f(0) = 0 $$

Now evaluate \(g(0)\) from the graph of \(y = g(x)\):

$$ g(0) = 4 $$

Thus:

$$ (g \circ f)(0) = 4 $$

Evaluate (f o g)(-1)

Using the Composite Functions and Graphical Function Evaluation knowledge points

$$ (f \circ g)(-1) = f(g(-1)) $$

From the graph of \(y = g(x)\):

$$ g(-1) = 5 $$

Now evaluate \(f(5)\) from the graph of \(y = f(x)\):

$$ f(5) = 1 $$

Thus:

$$ (f \circ g)(-1) = 1 $$

Evaluate (f o g)(4)

Using the Composite Functions and Graphical Function Evaluation knowledge points

$$ (f \circ g)(4) = f(g(4)) $$

From the graph of \(y = g(x)\):

$$ g(4) = 3 $$

Now evaluate \(f(3)\) from the graph of \(y = f(x)\):

$$ f(3) = -1 $$

Thus:

$$ (f \circ g)(4) = -1 $$

Answer:

(a) \((g \circ f)(-1) = 3\)

(b) \((g \circ f)(0) = 4\)

(c) \((f \circ g)(-1) = 1\)

(d) \((f \circ g)(4) = -1\)